Forecasting Volatility of Dutch Stocks: A Comparative Study on the Forecasting Performance of Option Implied Volatility and ARCH-type Models.

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Forecasting Volatility of Dutch Stocks:

A Comparative Study on the Forecasting Performance of Option

Implied Volatility and ARCH-type Models.

R.W. Groote Wolthaar S2317486 MSc. Finance

Faculty of Economics & Business University of Groningen

Supervisor: Dr. T. Boot January 2018

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List of Abbreviations

Abbreviation Full form Description/Note

ARCH Autoregressive Conditional

Heteroskedasticity

BS-Model Black-Scholes Model Option valuation model from which

the implied volatility is derived DM-test/statistic Diebold-Mariano test / statistic Test whether the difference

between forecasts is significant

GARCH Generalized Autoregressive

Conditional Heteroskedasticity

Type of ARCH model used to forecast volatility

GJR-GARCH The Glosten, Jagannathan, Runkle

GARCH

A GARCH model which includes an asymmetry term

h-horizon h-step ahead forecast horizon Forecasts the volatility h steps ahead

IV (model) Implied Volatility (Model) Used interchangeably with OIV

JB-test Jarque-Bera test Statistic used to test for normality

k-horizon k-period forecast horizon Forecast of the volatility for the

next k-(trading)day period

LTA Long-Term Average The long-term average of the range

based volatility using a rolling window.

MSE Mean Squared Error A quadratic loss function to

evaluate a models forecast OIV (model) Option Implied Volatility (Model) Used interchangeably with IV

VaR Value-at-Risk A common risk measure using

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Figures & Tables

Figure 1 – Distribution of Returns... 14

Figure 2 – Rolling Window ... 19

Figure 3 – IV One Day Ahead – Wolters Kluwer & Kon. Wessanen ... 22

Figure 4 - Fugro - One Day Ahead Forecasts ... 23

Figure 5 - One-Day Ahead Forecasts AEX Index ... 29

Table 1 – Descriptive Statistics Daily Returns (Jan 2009 – Jun 2017) ... 13

Table 2 – Mean Squared Errors ... 24

Table 3 – Model Performance One-Step Ahead Forecasts ... 25

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1. Introduction

Forecasting the volatility of a financial time-series has important applications in fields like risk management, derivatives pricing, market making, and asset pricing. Because of its importance and wide-usage, volatility forecasting has become an extensive field of research. Yet, there seem to be ambiguous results as to which model provides the most accurate forecasts (see e.g. Pilbean and Langeland, 2015; Canina and Figlewski, 1993). The aim of this paper will be to conduct a comparative study on the forecasting ability of several of these models. It will do so by conducting an out-of-sample forecast of the volatility of 26 Dutch listed equities.

The generalized autoregressive conditional heteroskedasticity (GARCH) model is a commonly used model in forecasting volatility which accounts for volatility clustering (Figlewski, 1997). Previous research shows that the GARCH model has prominent forecasting abilities (see e.g. Hansen and Lunde, 2005; Bentes, 2005; Pilbeam and Langeland, 2015). As such it is included in the study.

However, the standard GARCH model assumes a symmetric response to volatility, meaning that it assumes that positive and negative shocks of the same magnitude have similar effects on the volatility. In reality this is unlikely to be the case, as the leverage effect causes negative shocks to affect volatility more than positive ones (Andersen, Bollerslev, Christoffersen, and Diebold, 2006). To account for this asymmetry, the GJR-GARCH will be the second model included in this paper. The GJR-GARCH extends the regular GARCH by including a term which captures the asymmetric behaviour of the volatility (Glosten, Jogannathan, and Runkle, 1993)

Lastly, the option implied volatility (OIV) model is included. The implied volatility is the value of the volatility variable in the Black-Scholes option pricing model for which the derived option value equals the current market price. Compared to the ARCH-type models the OIV model might contain additional information, as its price contains the market’s expectations. Therefore, the option implied volatility model is said to be forward looking as opposed to the backwards looking ARCH models (Hull, 2006). Based on this we can thus derive the following research question:

Does option implied volatility perform better than ARCH-type models in forecasting volatility of Dutch listed equites?

The corresponding hypotheses are derived and formally stated in the literature review.

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rolling window, meaning that the first observation is dropped from the initial dataset and the

realized value of day t is added to conduct the forecast of day t+1. Out-of-sample forecasts were

conducted for the period 01-01-2016 till 30-06-2017 (384 trading days).

As volatility is a latent variable and thus cannot be observed, the range based volatility is used as a proxy. This is a volatility measure based on the intraday highs and lows introduced by Parkinson (1980). Subsequently, the out-of-sample forecasts are evaluated by taking the Mean Squared Errors between the observed range based volatility and the forecasted values.

The results show that based on the mean squared errors, the option implied volatility model outperforms the ARCH-type models. These results are significant according to the DM-test. In addition, no conclusive answer is found whether the GJR-GARCH outperforms the GARCH. When splitting up the Mean Squared Errors in the variance and bias, one can observe that while the variances of the different models are very similar, the biases are not. The results show that while the option implied volatility model is nearly an unbiased estimator of future volatility, the GARCH and GJR-GARCH models contain large upward biases.

The implied volatility models’ outperformance over the ARCH-type models suggest that option prices tend to include valuable information about the underlying asset. This is interesting as Canina and Figlewski (1993) found almost no correlation between implied volatility and future realized volatility on the S&P100 index, indicating that implied volatility does not contain additional information. Suggesting, that there could be either differences between indices and individual stocks, or that there are regional differences.

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2. Literature Review

2.1. Volatility

Volatility is a measure of risk that looks at the dispersion in the price of a financial instrument. More specifically, volatility is defined by Andersen et al. (2006) as the variability of a latent and random (stochastic) component of a time-series. Therefore, volatility is generally viewed as a type of uncertainty or risk, as an increase in volatility signifies an increase in the random component of the time-series.

It is generally assumed that volatility on equity markets is caused by trading based on the arrival of new information. As market participants update their valuations based on this information the prices of the equities change, resulting in volatility. However, this is not fully supported by empirical evidence. Research shows that only a quarter of the annual return volatility is justified by information about future earnings (see e.g., Campbell & Shiller, 1988; Shiller, 2000). It is more likely that a large part of volatility is caused by trading itself and its resulting noise (French and Roll, 1986; Black, 1986).

The turbulent nature of volatility makes forecasting difficult. Yet, accurate volatility forecasting methods are essential to financial economists as its usage is extensive. In risk management volatility is used to determine the Value-at-Risk (VaR) and the Expected Shortfall (ES), two of the main measures of market risk (see e.g. Angelidis, Benos, and Degiannakis, 2003; Hull, 2015). Speculators and hedgers in financial derivatives want to know the expected volatility during the duration of a financial instrument. Additionally, market makers might want to adjust the bid-ask spread on the basis of volatility expectations. Moreover, institutional investors incorporate expectations about future volatility in the selection of assets for their portfolio composition.

2.2. Generalized Autoregressive Conditional Heteroskedasticity

Mandelbrot (1963) found that there is a persistence of volatility shocks, meaning that returns volatility in the present day affects the unconditional variance of future volatility. This persistence in the volatility gives rise to volatility clustering: the tendency for financial returns to be clustered, i.e. periods of large absolute returns are followed by periods of large absolute returns, and periods of small absolute returns are followed by periods of small absolute returns (Brooks, 2014).

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allow for a heteroskedastic conditional volatility (Brooks, 2014), meaning that the conditional volatility may change over time, but the unconditional volatility does not.

What is described up to now is the basic ARCH model by Engle (1982). The Generalized ARCH (GARCH) process extends the ARCH by adding the lagged conditional variance to model the error variance. As such, the error variance will follow an autoregressive moving average model (ARMA), instead of a standard autoregressive model (Bollerslev, 1986). By lagging the conditional variance the GARCH is essentially an ARCH(∞) process.

Over time the forecasts of GARCH-type models will move towards the unconditional variance of the time series. This is a favourable characteristic in volatility forecasting as volatility tends to show mean-reverting behaviour (Engle and Patton, 2001).

2.3. Asymmetric Volatility

A limitation of the GARCH model is that it enforces a symmetric response to volatility. Whereas, according to the leverage effect, a negative shock is likely to have more effect than a positive shock. Additionally, asymmetry also causes the volatility feedback effect, in which “heightened volatility requires an increase in the future expected returns to compensate for the increased risk, in turn necessitating a drop in the current price to go along with the initial increase in volatility”. (Andersen, Bollerslev, Christoffersen, Diebold, 2006, pp. 803).

Hansen and Lunde (2005) found no evidence that any of the 330 ARCH-family models they tested outperformed the GARCH(1,1) in forecasting the out-of-sample volatility of the DM/$ exchange rate and IBM stock. On the contrary, Awartani and Corradi (2005) found different results. In predicting the volatility of the S&P500 they found that the asymmetric models are more effective than the GARCH method for multiple forecasting horizons.

To account for this asymmetry term, the asymmetric GJR-GARCH model will be included in this study. The Glosten, Jagannathan, Runkle-GARCH (1993) model extends the GARCH to include an indicator function that captures the effects caused by the asymmetry. Based on asymmetry term in the GJR-GARCH model the first hypothesis can be derived:

H0A: The GJR-GARCH model does not outperform the regular GARCH model in

forecasting out-of-sample volatility

H1A: The GJR-GARCH model outperforms the regular GARCH model in forecasting

out-of-sample volatility. 2.4. Option Implied Volatility

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the risk free rate; and the volatility of the underlying asset.1 Of these five variables the volatility of the underlying asset is the only variable that cannot be observed directly, but it can be derived. Given that the other four variables are known, the Implied volatility is the value of the volatility variable in the Black-Scholes model for which the derived option value equals the current market price (Figlewski, 1997).

For any given underlying asset, there exist many options with varying time-to-maturities and strike prices. If the Black-Scholes would hold, these options would have to be priced such that their implied volatilities are equal (Mayhew, 1995), this is generally not the case2. The implied volatilities of the different options (with the same underlying asset) tend to differ across strike price (the ‘volatility skew’) and across time-to-maturity (the ‘term structure of volatility’) (Hull, 2006). Rubinstein (1985) and Sheikh (1991) investigate these difference of option implied volatility across strike prices and time-to-maturity by investigating options traded on the CBOE and the S&P100, respectively. Both find differences in implied volatility across strike prices and time-to-maturity.

A possible explanation for volatility skew has to do with leverage. When a company’s equity increases in value, the leverage of the firm decreases (since debt remains constant). As the leverage decreases, the company becomes less risky, which decreases volatility (Hull, 2006). The opposite is expected when equity decreases in value.

Another explanation is what Rubinstein (1994) describes as “crash-o-phobia”, which describes the phenomena that ever since Black Monday (the 1987 flash crash) there is a higher demand for out-of-the-money options, likely to protect against extreme events. This is investigated further by Jackwerth and Rubinstein (1996) who looked at European options of the S&P500 index. They found that prior to the flash crash implied volatility was less dependent on strike price, to such an extent that the volatility skew was generally non-existent prior to October 1987.

Thus, now that we have concluded that there are different implied volatilities and we know what is likely to cause them, which one to choose? Mayhew (1995) conducted a literature review on implied volatility and found that the consensus is that the Black-Scholes model performs best for at-the-money options with a time-to-maturity of 1-2 months. This is due to the fact that for at-the-money options, the strike price is equal to the price of the underlying asset and thus not affected by the premium for the ‘crash-o-phobia’ or the leverage effect. This paper accounts for this in the data selection (see section 3.2.).

1_{The Black-Scholes formula is given in appendix 1.}

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Compared to the model based forecasts, implied volatility might have additional informational value. The reason is that, according to the (semi-)strong form of the efficient market hypothesis, market prices include all (public) information. As such, the implied volatility includes the markets expectations and is thus forward looking, whereas the historical models are backward looking (Hull, 2006).

Nonetheless, one should note that implied volatility is likely to be biased in predicting volatility since market prices are affected by various other factors that are not taken into account in the BS model, such as liquidity (Bentes, 2015) and the bid-ask spreads (Jorion, 1995). Moreover, included in the option implied volatility is the volatility premium, which is the premia to compensate for the risk investors bear that relate to sudden sharp changes in the market volatility (Lombardi and Schrimpf, 2014; Rennison and Pedersen, 2012).

Due to the volatility premium the option implied volatility is generally slightly higher than the realized volatility. This can also be observed from the implied volatility of Wolters Kluwer in appendix 8.

Based on the fact that the option implied volatility model is forward looking in nature, and thus is likely to incorporate more information, the following hypothesis can be derived:

H0B: The option implied volatility model does not outperform the ARCH-type models in

forecasting out-of-sample volatility.

H1B: The option implied volatility model outperforms the ARCH-type models in

forecasting out-of-sample volatility. 2.6. Current Literature

When looking at previous research on these different methods one finds contrasting results as to which preforms best. Pilbeam and Langeland (2015) show that implied volatility outperforms (three different) GARCH-family models on four different foreign exchange markets. Similar results were found by Christensen and Prabhala (1998) by investigating the S&P100 index. In addition, Pong, Shackleton, Taylor, and Xu (2004) conducted out-of-sample forecasts of three different currency exchange rates (USD/GBP, USD/DEM, and USD/JPY). They found that the implied volatility outperforms the GARCH model in forecasting the volatility on longer horizons.

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This paper will add to the current literature by using a new dataset to test whether or not the implied volatility performs better than the (GJR-)GARCH models. Additionally, current research tends to either focus on American indices (e.g. Koopman, Jungbacker, and Hol (2005); Christensen and Prabhala (1998); Hansen and Lunde (2005)), on the indices of developing countries (e.g. Bentes (2005)) or various foreign exchange markets (Pong, Shackleton, and Taylor (2004); Pilbeam and Langeland (2015). Far less research has been done on the volatility of (Western-) European equities, this paper will add to this literature by investigating Dutch listed companies.

2.7. Market Differences

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3. Data

The sample of companies started off as a list of all publicly traded companies listed on the Euronext Amsterdam. The list was then filtered down by excluding companies for which price and/or option data was unavailable on DataStream for the period January 2009 up to (and including) June 2017. This resulted in a sample of 26 companies. For these companies the following data was collected. It should be noted that the excluding companies for which no option data is available favours the implied volatility model, as companies for which the data is unavailable tend to be companies with inactive option markets.

3.1. Price Data

Firstly, the daily closing prices were collected from DataStream for the 26 companies in the sample. The daily return indexes (DataStream code “RI”) are used since these are adjusted closing prices corrected for stock splits and dividends. The adjustment for stock splits is needed since while the price of the stock will change due to a split, it’s value will not. Therefore, the adjustment is needed to correctly calculate the historical returns. The same applies to the dividend adjustment. However, dividends do reduce a company’s value (by the amount of the dividend), but this is paid out to the investor and thus does not affect the investors’ return. From the return index, the daily returns for company i, can be derived as:

𝑅𝑖,𝑡 = ln ( 𝑅𝐼_𝑖,𝑡 𝑅𝐼𝑖,𝑡−1

) (1)

Where, RIi,t and RIi,t-1 denote the return index for stock i at day t, and the return index for stock i at day t-1, respectively. The daily returns will be used as inputs to determine the volatility using the GARCH and GJR-GARCH models.

Secondly, the realized volatility is needed to test the accuracy of the models. However, as discussed in the literature review, volatility cannot be directly observed. Consequently, a range-based volatility estimator is used, for which the intraday lows and high were collected from DataStream. The reasoning behind choosing the range-based volatility and its implications are discussed in the research methodology (section 4.3).

3.2. Option Data

The daily implied volatilities for the companies in the sample were obtained directly via DataStream. These values are derived from European call and put options using the Black-Scholes model. DataStream calculates the implied volatility by using the nearest two at-the-money options (one above and one below the underlying price)3, this is in line with Mayhew’s

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(1995) recommendation. Since the implied volatilities are denoted in yearly values, we transform them in daily implied volatilities by dividing the values by the square root of the number of trading days in a year (256 days on the Euronext Amsterdam).

Normally when forecasting volatility based on implied volatility, equities with a sufficiently active option market are preferred as illiquid options are less efficiently priced. However, no equities were excluded in this paper based on the activity of its option market. Rather, the daily trading volume of option contracts have been collected, which will act as a proxy for liquidity. As such, a comparison can be made based on the level of liquidity of an option and the effectiveness of the option’s implied volatility in forecasting the volatility of the underlying asset.

3.3. Dataset

After removing non-trading days (like holidays) we are left with a sample of 2177 observations per company. This dataset will be split up in a training set and a testing set, such that an out of sample forecast can be performed. The split date will be at 01-01-2016, resulting in a training period from 01-01-2009 till 31-12-2015 containing 1793 observations per company. The test set (i.e. the days that will be forecasted) from 01-01-2016 till 30-06-2017 contains 384 observations per company.

As the objective of this paper is to forecast volatility, including its spikes, outliers will not be removed from the sample. Having said that, an exception was made for the implied volatility of Wolters Kluwer, which clearly contained an error on the 11th of September 2011 when implied (daily) volatility moved from ~2% to ~50% for no reason. This data point was consequently removed from the sample (see appendix 8 for the distribution before and after the removal).

3.4. Descriptive Statistics

Although we split up our dataset in the previous sample, this section will look at the descriptive statistics of the entire sample. By looking at the mean and median values in table 1, it can be observed that all distributions are similarly centred. However, as indicated by the minimum and maximum values the time series are differently distributed.

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From figure 1 it can already be observed that Heineken has a lower historical volatility than Binck and Fugro. Note how, while the mean return is identical, the occurrence of large high/low returns is more likely for Binck and Fugro. This can also be observed from the standard deviation in table 1.

Table 1 – Descriptive Statistics Daily Returns (Jan 2009 – Jun 2017)

The table contains descriptive statics of the daily returns for all 26 companies in the dataset, for which descriptive statistics are calculated over the entire period (i.e. contains the training and test set). Kurtosis denotes excess kurtosis. The abbreviation JB is used for the Jarque-Bera statistic, JB stat is the corresponding p-value of the Jarque-Bera test. Number of observations for each company is 2177.

To further examine the distributions we can investigate the tails by looking at the kurtosis. As table 1 shows, all distributions show excess kurtosis. Appendix 7 shows the distributions of Heineken (the company with the lowest excess kurtosis) and Fugro (the highest excess kurtosis in our sample). These leptokurtic distributions (likely a result of volatility clustering) are not problematic for GARCH modelling (Brooks, 2014). On the contrary, Angelidis, Benos, and Degiannakis (2003) found when using GARCH models to forecast volatility for Value at Risk (VaR) estimation, the leptokurtic distributions produced better one-step ahead VaR forecasts. Using the skewness and excess kurtosis, the Jarque-Bera test can be conducted, which tests whether the returns follow a normal distribution. As can be seen from table 1 the aforementioned leptokurtosis and skewness result in high values for the Jarque-Bera statistic. As a result, the tests’ null hypotheses of normally distributed returns are rejected at the 1% level

Average Median Max Min St. dev Skewness Kurtosis JB JB stat

ROYAL DUTCH SHELL A 0.000 0.000 0.065 -0.080 0.014 -0.197 3.595 47 0.000

UNILEVER DR 0.001 0.000 0.124 -0.065 0.013 0.477 6.680 1335 0.000

HEINEKEN 0.001 0.000 0.080 -0.079 0.013 0.082 3.211 7 0.037

PHILIPS ELTN 0.001 0.000 0.093 -0.092 0.018 0.042 2.715 8 0.017

KON. AHOLD DELHAIZE 0.000 0.000 0.063 -0.100 0.013 -0.296 4.882 360 0.000

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and non-normality should be assumed (with an exception for the tests of Heineken and Phillips, which are rejected at the 5% level).

Figure 1 – Distribution of Returns

The figures show the distributions of Heineken, Binck, and Fugro. Whereas the left figure shows the entire distribution the right figure cuts of the x-axis at -0.10 and +0.10.

As can be observed from appendix 7 (and figure 2) the data of 2009 appears to have significantly higher volatility. If we split the data up, as shown in appendix 4, one can clearly see that this is the case for most companies. Not only does the data show that almost for every company the standard deviation is higher in 2009 than in 2010 and in the entire sample excluding 2009. But this can also be observed from the range (the difference between the maximum and minimum value). The differences appear to be the largest for companies in the financial industry. This paper accounts for this difference by conducting forecasts with both a rolling window and a recursive window. This will be further discussed in section 4.5.

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4. Research Methodology

4.1. GARCH Model

The GARCH and GJR-GARCH models4 in this paper follow the process as described by Andersen et al. (2006). The parameters can be estimated by the maximum likelihood estimation as described by Hull (2006). The return process is given by:

𝛾𝑡= 𝜇𝑡|𝑡−1+ 𝜎𝑡|𝑡−1𝑧𝑡, 𝑧𝑡 ~ 𝑖. 𝑖. 𝑑., 𝐸(𝑧𝑡) = 0, 𝑉𝑎𝑟(𝑧𝑡) = 1 (2) Where μt|t-1 is the conditional mean and σt|t-1 the conditional variance of the return process given all available information at time t-1. The GARCH(1, 1) model for the conditional variance is then defined as:

𝜎_{𝑡|𝑡−1}2 = 𝜔 + 𝛼𝜖_𝑡−12 + 𝛽𝜎_{𝑡−1|𝑡−2}2 , 𝜖_𝑡−1≡ 𝜎_{𝑡−1|𝑡−2}𝑧_𝑡−1, 𝜔 = 𝛾𝜎2 (3) Where, ω > 0, α ≥ 0, and β ≥ 0 to ensure that the conditional variance remains positive. Then, using the maximum likelihood method, the parameters α, β, and γ are set such that they maximize: ∏ [ 1 √2𝜋𝜎_{𝑡|𝑡−1}2 exp (−𝜇𝑡|𝑡−1 2 2𝜎_{𝑡|𝑡−1}2 ) ] 𝑛 𝑡=1 (4)

By taking the logarithms of (4) the expression can be rewritten as:

∑ [− ln(𝜎_{𝑡|𝑡−1}2 ) − 𝜇𝑡|𝑡−1 2 𝜎_{𝑡|𝑡−1}2 ] 𝑚 𝑡=1 (5)

The long-term variance, σ2 , can be derived from the expected variance at a given period t. The expected variance for period t|t-1 is given by:

𝐸[𝜎_{𝑡|𝑡−1}2 ] = 𝛾𝜎2_{+ 𝛼𝐸[𝜖}

𝑡−12 ] + 𝛽𝐸[𝜎𝑡−1|𝑡−22 ] (6)

Where, 𝐸[𝜖_𝑡−12 ] = 𝐸[𝜎_{𝑡−1|𝑡−2}2 𝑧_𝑡−12 ] = 𝐸[𝜎_{𝑡−1|𝑡−2}2 ]𝐸[𝑧_𝑡−12 ] = 𝐸[𝜎_{𝑡−1|𝑡−2}2 ], as such (6) can be rewritten as:

𝐸[𝜎_{𝑡|𝑡−1}2 ] = 𝛾𝜎2_{+ 𝛼𝐸[𝜎}

𝑡−1|𝑡−22 ] + 𝛽𝐸[𝜎𝑡−1|𝑡−22 ] (7) Under stationarity, that is 𝐸[𝜎_{𝑡|𝑡−1}2 ] = 𝜎2, the weights γ, α, and β need to sum up to unity for (6) to hold. Therefore, the long run variance (or unconditional variance) can be derived as:

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𝜎2 = 𝜔(1 − 𝛼 − 𝛽)−1 (8)

The h-step ahead forecast can now be expressed as: 𝜎_{𝑡+ℎ|𝑡}2 = 𝜎2 _{+ (𝛼 + 𝛽)}ℎ−1_(𝜎

𝑡+1|𝑡2 − 𝜎2) (9)

Thus, as the forecast horizon approaches infinity, the forecasted variance approaches the long-term (unconditional) variance of our return series, i.e. 𝜎_{𝑡+ℎ|𝑡}2 → 𝜎2 𝑎𝑠 ℎ → ∞. The h-step ahead volatility can then be defined as the square root of the variance calculated in (9).

Lastly the k-period variance forecast can be expressed as: 𝜎_{𝑡:𝑡+𝑘|𝑡}2 = 𝑘𝜎2_{+ (𝜎}

𝑡+1|𝑡2 − 𝜎2)(1 − (𝛼 + 𝛽)𝑘)(1 − 𝛼 − 𝛽)−1 (10) As k increases the second term in (10) approaches zero, such that the forecasted standardized volatility per day (𝑘−1𝜎𝑡:𝑡+𝑘|𝑡2 , ) approaches the unconditional variance. Appendix 10 shows this for the forecasts of Ordina using k = 1, 5, and 20 horizons.

4.2. GJR-GARCH Model

The regular GARCH model discussed in the previous section assumes that positive and negative shocks of the same magnitude have an identical effect on future volatility. The GJR-GARCH is an extension on the GARCH with an extra term that accounts for the asymmetry effect (Andersen et al., 2006). The GARCH model from (3) can simply be transformed to include the asymmetry term. As such, the GJR-GARCH model can be expressed as:

𝜎_{𝑡|𝑡−1}2 = 𝜔 + 𝛼𝜖_𝑡−12 + 𝛾𝜖_𝑡−12 𝐼(𝜖𝑡−1< 0) + 𝛽𝜎𝑡−1|𝑡−22 (11) Where I(∙) denotes the indicator function, which equals one if ϵt-1 < 0, and zero otherwise. Such that, when γ is positive, a negative volatility shock has a larger impact on the variance than a positive shock of the same magnitude would. Since zt follows an independent identically distribution, we can assume that:

𝑝(𝑧_𝑡≡ 𝜎_{𝑡|𝑡−1}−1 𝜖_𝑡< 0) = 0.5 (12)

Equation (9) can now be adjusted for the asymmetry term by incorporating (11) and (12). Such that the h-step ahead forecast using the GJR-GARCH method can be expressed as:

𝜎_{𝑡+ℎ|𝑡}2 = 𝜎2_{+ (𝛼 + 0.5𝛾 + 𝛽)}ℎ−1_(𝜎

𝑡+1|𝑡2 − 𝜎2) (13)

Where the long-run variance equals:

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The h-step ahead volatility for the GJR-GARCH is not discussed in Andersen et al. (2006), therefore we derive it in similar fashion as (10). Essentially, (10) can be derived from (9) by multiplying the long-term (unconditional) variance by k-periods and taking a geometric series, with size k, from the second part of the equation. Hence, for our GJR-GARCH model a similar geometric series can be obtained for the second part of equation (13), i.e.:

(𝜎_𝑡+1|𝑡2 − 𝜎2_{) + (𝜎} 𝑡+1|𝑡2 − 𝜎2)(𝛼 + 0.5𝛾 + 𝛽) + (𝜎𝑡+1|𝑡2 − 𝜎2)(𝛼 + 0.5𝛾 + 𝛽)2+ (𝜎_𝑡+1|𝑡2 − 𝜎2_{)(𝛼 + 0.5𝛾 + 𝛽)}3_{+ ⋯ + (𝜎} 𝑡+1|𝑡2 − 𝜎2)(𝛼 + 0.5𝛾 + 𝛽)𝑘−1 = ∑(𝜎_𝑡+1|𝑡2 − 𝜎2_{)(𝛼 + 0.5𝛾 + 𝛽)}𝑘 𝑘−1 𝑘=0 = (𝜎_𝑡+1|𝑡2 − 𝜎2₎1 − (𝛼 + 0.5𝛾 + 𝛽) 𝑘 (1 − 𝛼 − 0.5𝛾 − 𝛽) (15)

We can now add (15) to the k-multiple of the long-term variance to obtain the k-period variance under the GJR-GARCH method:

𝜎_{𝑡:𝑡+𝑘|𝑡}2 = 𝑘𝜎2 _{+ (𝜎}

𝑡+1|𝑡2 − 𝜎2)(1 − (𝛼 + 0.5𝛾 + 𝛽)𝑘)(1 − 𝛼 − 0.5𝛾 − 𝛽)−1 (16)

4.3. Range Based Volatility

Unfortunately, volatility is a latent variable and it thus needs to be inferred ex-post. The most common method is to use absolute daily close-to-close returns, but since it only incorporates closing prices it’s performance is inferior compared to other measures. Andersen and Bollerslev (1998) show that volatility forecasting models perform worse when the daily squared returns are used as a proxy for volatility and show that intraday (high-frequency) volatility is a more efficient ex-post estimator.

However, due to the bid-ask spread the observed intraday (high-frequency) prices contain noise, since the observed price equals the true price plus half of the bid-ask spread. This spread causes transaction to bounce between bid and ask prices, which consequently increases the realized volatility. Over time this bounce will result in a large cumulated bias (Alizadeh, Brandt, and Diebold, 2002).

The range based volatility by Parkinson (1980) is an extreme value estimator based on the intraday low and high, defined as:

𝑟𝑎𝑛𝑔𝑒 𝜎𝑡 = √ 1 4 ln 2(ln 𝐻_𝑡 𝐿𝑡 ) 2 (17)

Additionally, we can calculate the volatility of a k-period horizon as:

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Where Ht denotes the intraday high and Lt the intraday low at day t. As the bid-ask bounce for the range-based only equals the average bid-ask spread (i.e. two times half the spread), its less likely to be seriously affected by it (Alizadeh et al., 2002).

A small downside of range based volatility is that it does not take into account the volatility during non-trading hours. However, asset prices are much more volatile during exchange trading hours than during non-trading hours (French & Roll, 1986; Black, 1986). In all, though, range based volatility is a well-suited volatility measure.

4.4. Implied Volatility Model

In the implied volatility model, the range based volatility will be estimated and forecasted using the following linear model:

𝑅𝑉_𝑡+ℎ = 𝛼 + 𝛽𝐼𝑉_𝑡+ 𝜖_𝑡 (19)

While theoretically one would expect α =0 and β = 1, it is more likely that alpha will be around zero and beta slightly smaller than one. This is due to the fact that the volatility premium causes option implied volatility to be a slightly biased estimator. As the volatility premium increases the value of the implied volatility the estimator is expected to be upward biased.

Additionally, since the variance of the return increases linearly over time (Hull, 2015), the volatility increases with the square root of time. As such, we can calculate the implied volatility of a k-period horizon as:

𝐼𝑉_{𝑡:𝑡+𝑘|𝑡} = 𝐼𝑉_𝑡√𝑘 (20)

4.5. Forecasting Window and Horizon

The out-of-sample volatility forecasts are conducted using a daily rolling window. This implies that while the first forecast at day t is made used on parameters obtained from the entire in-sample dataset (i.e. 01/01/2009 – 31/12/2015), for subsequent forecast the window shifts by one day. Thus, the forecasts performed at day t+1 are based on the parameters obtained over the period 02/01/2009 – 01/01/2016.

As discussed in the data section, the volatility was significantly higher in 2009 than in succeeding periods (the two most extreme cases are visualized in figure 2). With a rolling window, however, these periods will eventually drop from the training data. For example, the training period for the forecast of 30-06-2017, the last forecast in the sample, completely excludes 2009 (and parts of 2010). As shown by the grey area in figure 2.

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(like the rolling window), but over time increases with each new observation. For clarity all figures, tables, and results mentioned are based on the rolling windows unless it is explicitly specified that it is based on a recursive window.

As discussed in the literature review there are different uses for volatility forecasts. Therefore, there might be a need for different forecasting horizons. For example, a risk manager that wants to calculate the one day VaR needs a different forecasting horizon than a derivatives trader whom wishes to trade a contract with a remaining life of one month.

Figure 2 – Rolling Window

The figures show the realized range based volatility of ING Groep and AEGON. The shaded area corresponds to the rolling window used to forecast the last observation in our sample.

In addition, one-, five- and twenty-step ahead forecast are conducted, such that there are forecast for the next trading day, one week later and four weeks later. To distinguish between these two types of horizons, this paper uses the variable h to denote h-step ahead forecasts, and the variable k to denote k-period forecasts.

4.6. Forecast Evaluation

To evaluate the effectiveness of the different forecasting methods the mean squared errors are calculated. The mean squared error is chosen since it’s a quadratic loss function and will thus penalize larger forecast errors (Brooks, 2014). The MSEs over the sample period T are calculated as followed: 𝑀𝑆𝐸̂_𝑇,𝑖𝑀 = 1 𝑛∑(𝑌̂𝑡− 𝑌𝑡) 2 𝑛 𝑡=1 (21)

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The MSE consists of two components, the variability of the estimator and its bias. The variability can be seen as the precision of the estimator, which is the variation of the difference between the forecasted and observed values. The bias measures the accuracy, which is the mean of the difference between the forecasted values and the observed values (Brooks, 2014). Thus, when the estimator is unbiased, 𝐸(𝑌̂) = 𝑌, the MSE equals its variance. The relationship between variance and bias can be more formally specified as:

𝑀𝑆𝐸 = 𝑉𝑎𝑟(𝑌̂) + [𝐸(𝑌̂) − 𝑌]2 = 𝑉𝑎𝑟(𝑌̂) + 𝐵𝑖𝑎𝑠(𝑌̂)2 (22)

The fact that one model might perform better in our sample does not mean it would perform better for the entire population. Let’s say one tests the GARCH and GJR-GARCH for ING Groep over period T and finds 𝑀𝑆𝐸̂_{𝑇,𝐼𝑁𝐺}𝐺𝐽𝑅 < 𝑀𝑆𝐸̂_{𝑇,𝐼𝑁𝐺}𝐺𝐴𝑅𝐶𝐻, this does not necessarily imply that 𝑀𝑆𝐸_𝐼𝑁𝐺𝐺𝐽𝑅 < 𝑀𝑆𝐸_𝐼𝑁𝐺𝐺𝐴𝑅𝐶𝐻. The Diebold-Mariano (1995) test can be used to test whether the GJR is actually more efficient in this example, or if the outperformance could have occurred by chance.

Specifically, the Diebold-Mariano tests whether the predictive accuracy of two forecasting models is equal. More formally, it tests the null hypothesis that the expected losses of both models are equal, that is 𝐻0: 𝑀𝑆𝐸_𝑇,𝑖1 = 𝑀𝑆𝐸_𝑇,𝑖2 , against 𝐻𝐴: 𝑀𝑆𝐸_𝑇,𝑖1 ≠ 𝑀𝑆𝐸_𝑇,𝑖2 (Diebold, 2017). The test statistic can be calculated as:

𝐷𝑀_1,2 = 𝑑̅1,2 𝜎̂_𝑑̅_1,2

𝑑

→ 𝑁(0,1) (23)

Where 𝜎̂_𝑑̅_1,2 is a consistent estimate of the standard deviation of 𝑑̅_1,2 , and 𝑑̅_1,2 is the sample mean loss differential, i.e.:

𝑑̅_1,2 = 1

𝑇∑ 𝑑1,2,𝑡 𝑇

𝑡=1 (24)

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5. Results

From the mean squared errors and their significance test (shown in table 2 and 3, respectively) we can conclude that the option implied volatility significantly outperforms both the GARCH and GJR-GARCH model for all 26 companies in the sample. In addition, no decisive evidence was found that the GJR-GARCH outperforms the GARCH model. Regrettably, neither did we find any evidence that the GARCH models outperformed the long-term historical average of the volatility. Moreover, what is alarming is that the contrary is the case; the long-term average predominantly outperforms both GARCH models.

The following subsections will first discuss the results in more depth and will then look at possible explanations for these results.

5.1. Implied Volatility

The definite winner of the out-of-sample forecasts conducted in this paper is the implied volatility model. From table 2 we can observe that the IV model outperforms the GARCH and GJR-GARCH for all companies and on all horizons. Moreover, the DM-test statistics in appendix 17 show that nearly all of these differences in performance are statistically significant. Based on this we reject the null hypothesis (H0B) that the OIV model does not outperform the ARCH-type models. This is similar to the findings of Pilbeam and Langeland (2015) and Christensen and Prabhala (1998). In addition, the fact that the implied volatility outperforms the ARCH-type models on longer forecasting horizons is in line with Pong, Shackleton, Taylor, and Xu (2004)

The beta’s in our estimated models tend to be below one, which is partly explained by the volatility premium. As a result, the observed ex-post implied volatility to be slightly higher than the forecasted implied volatility. This phenomena is visualized in appendix 12, showing the one-step ahead forecast for Fugro and includes both the observed and forecasted implied volatility. The graph shows that there is a small, yet consistent difference between the observed and forecasted value.

What is peculiar is that for some companies with a relatively low beta specified in their implied volatility model still significantly outperform the GARCH-type models (see e.g. Wereldhave, Kon. KPN, and Kon. Wessanen in Appendix 9). As beta — the slope of the regression —

indicates the historical relation between the implied volatility and the range based volatility, one would expect that a low beta results in low forecasting performance.

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actually close to the predicted values and show the movements in volatility. Whereas, the forecasts of Kon. Wessanen is essentially an long-term average with some minor movements. Moreover, when we look at the effect of the volume of options traded on the performance of the implied volatility (see appendix 22) there seems to be no clear relationship. This is peculiar as an active option market is necessary to fully benefit from the informational content in prices. A possible explanation could be that all companies’ options in our sample are active enough to fully convey the information.

Figure 3 – IV One Day Ahead – Wolters Kluwer & Kon. Wessanen

The first figure shows the forecasted implied volatility of Wolters Kluwer and the second one the implied volatility of Kon. Wessanen. The graphs show the difference between having a low alpha and high beta (Wolters Kluwer) and having a high alpha but low beta (Kon. Wessanen).

5.2. GJR-GARCH

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5 in favour of the GJR-GARCH. Indicating that while there is no definite answer whether the asymmetry term in the GJR-GARCH increases performance, we see that it some cases it might. Similar results are found when looking at forecasts using other horizons5_{. Based on the} indecisiveness of the results we fail to reject the null hypothesis (H0A) that the GJR-GARCH does not outperform the GARCH.

The fact that a conclusive answer can’t be drawn on the performance of the GJR-GARCH makes comparing it to the current literature difficult. Being pedantic we could say that Hansen and Lunde (2005) are right in the fact that there is no evidence that the GJR-GARCH outperforms the regular GARCH. Yet, that answer would seem incomplete, since it differs per company, a more correct statement would be that it depends. Although the data gives no clear indication on what could cause it.

5.3. GARCH

Both the GJR-GARCH as the GARCH perform significantly worse than the option implied volatility model for all companies in our sample. Since the option implied volatility is forward looking we initially expected it to perform better than the ARCH-type models, just not to this extent. Although the implied volatility performs relatively well, as can be seen in figure 4, the main cause of the large difference in performance is due to the poor performance of the GARCH models.

Figure 4 - Fugro - One Day Ahead Forecasts

The figure shows the one step ahead forecasts of the GARCH, GJR-GARCH and Implied Volatility models for the company Fugro. The range based volatility is used as a proxy for realized volatility and is the observed, ex-post value.

From figure 4 we can observe that volatility persists in the ARCH-type models for a long time. For example, when we look at either March 2016 or August 2016, we see that these shocks

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[24] Table 2 – Mean Squared Errors

Mean squared errors of the GARCH, GJR-GARCH (GJR), Option Implied Volatility (IV) and the Long Term Average (LTA) of the range based volatility (using a rolling window). One day period not displayed, since (by definition) it is identical to the one day ahead forecast.

GARCH GJR IV LTA GARCH GJR IV LTA GARCH GJR IV LTA GARCH GJR IV LTA GARCH GJR IV LTA

ROYAL DUTCH SHELL 0.381 0.334 0.295 0.550 0.414 0.351 0.318 0.543 0.433 0.343 0.387 0.408 0.940 0.730 0.631 2.034 3.006 1.950 1.274 5.990 UNILEVER 0.481 0.424 0.306 0.358 0.518 0.445 0.317 0.358 0.553 0.500 0.335 0.363 1.629 1.312 0.857 1.086 4.984 3.492 2.149 2.551 HEINEKEN 0.259 0.269 0.158 0.244 0.275 0.289 0.167 0.246 0.313 0.334 0.195 0.255 0.758 0.812 0.324 0.700 2.335 2.548 0.583 1.861 KONINKLIJKE PHILIPS 0.412 0.409 0.215 0.399 0.459 0.467 0.231 0.404 0.566 0.585 0.274 0.392 1.342 1.365 0.470 1.227 5.528 5.787 1.056 3.870 KON. AHOLD 0.409 0.450 0.340 0.374 0.453 0.490 0.363 0.373 0.424 0.423 0.373 0.358 1.334 1.533 1.070 1.169 2.407 2.356 1.945 2.104 AKZO NOBEL 0.591 0.545 0.291 0.424 0.620 0.593 0.318 0.425 0.723 0.730 0.376 0.428 1.916 1.732 0.664 1.179 6.252 5.874 1.992 2.807 AEGON 1.341 1.519 0.774 0.933 1.428 1.568 0.796 0.943 1.505 1.585 0.861 0.958 4.114 4.797 1.739 2.313 11.517 14.000 3.550 4.233 KON. BOSKALIS WEST 1.021 1.396 0.502 0.672 1.114 1.491 0.503 0.679 1.346 1.720 0.529 0.706 3.781 5.583 1.280 2.053 13.445 20.228 1.384 5.288 SBM OFFSHORE 1.277 1.278 0.541 0.820 1.342 1.369 0.580 0.818 1.449 1.525 0.652 0.702 4.315 4.303 1.123 2.468 15.827 16.447 2.170 6.900 KON. BAM GROEP 1.863 1.970 0.687 1.192 2.106 2.115 0.761 1.205 2.569 2.424 0.886 1.251 7.647 7.906 1.973 4.112 30.410 29.456 5.650 12.208 CORBION NV 0.945 0.953 0.719 0.815 0.976 1.017 0.719 0.812 1.064 1.170 0.774 0.830 3.010 3.159 2.096 2.458 5.853 6.611 3.069 3.647 WERELDHAVE 0.237 0.242 0.149 0.207 0.266 0.277 0.156 0.209 0.359 0.368 0.170 0.215 0.752 0.792 0.333 0.595 3.275 3.460 0.776 1.684 KON. WESSANEN 0.777 0.783 0.554 0.660 0.822 0.833 0.566 0.663 0.842 0.846 0.587 0.623 2.091 2.134 1.352 1.558 5.572 5.732 2.558 3.017 WOLTERS KLUWER 0.248 0.258 0.170 0.258 0.267 0.274 0.182 0.255 0.329 0.324 0.217 0.257 0.660 0.698 0.317 0.719 2.299 2.292 0.681 1.885 RELX NV 0.254 0.255 0.190 0.266 0.277 0.280 0.199 0.268 0.341 0.344 0.237 0.272 0.676 0.681 0.402 0.776 2.205 2.257 0.943 2.029 BINCKBANK 0.939 1.139 0.589 0.642 0.973 1.105 0.611 0.644 1.002 1.047 0.635 0.654 2.805 3.581 1.334 1.612 7.631 9.147 3.419 3.830 ORDINA 2.188 2.201 2.006 2.215 2.727 2.734 2.069 2.210 2.816 2.851 2.199 2.270 7.660 7.660 6.201 6.883 16.898 17.341 14.246 14.070 AALBERTS 0.394 0.472 0.277 0.678 0.430 0.515 0.296 0.680 0.580 0.686 0.360 0.700 1.284 1.652 0.623 2.473 5.073 6.744 1.401 8.073 KON. DSM 0.417 0.389 0.240 0.359 0.454 0.423 0.237 0.361 0.523 0.500 0.294 0.356 1.216 1.066 0.356 0.970 4.266 3.838 0.811 2.574 RANDSTAD HOLDING 0.803 0.994 0.470 0.656 0.865 1.041 0.484 0.662 1.032 1.159 0.555 0.678 2.535 3.385 1.142 1.868 9.470 12.237 3.287 5.211 ING GROEP 1.156 1.191 0.609 1.197 1.392 1.403 0.631 1.207 1.800 1.773 0.736 1.250 4.432 4.598 1.641 4.034 17.418 17.598 3.971 11.294 FUGRO 3.519 2.690 1.342 1.866 3.472 2.755 1.317 1.877 4.651 3.809 1.657 1.834 12.241 8.621 2.512 6.789 52.386 32.812 6.271 23.445 KON. KPN 0.558 0.565 0.386 0.455 0.591 0.588 0.386 0.457 0.652 0.642 0.421 0.453 1.613 1.621 0.911 1.207 4.274 4.166 1.570 2.226 ASML HOLDING 0.649 0.593 0.286 0.460 0.755 0.677 0.319 0.463 0.899 0.891 0.360 0.449 2.290 1.973 0.602 1.264 8.870 7.871 1.165 3.025 ASM INTERNATIONAL 0.688 0.746 0.466 0.715 0.743 0.807 0.484 0.721 0.855 0.913 0.519 0.744 1.952 2.268 1.053 1.915 5.627 6.748 2.029 4.458 KON. VOPAK 0.586 0.609 0.349 0.421 0.621 0.627 0.363 0.417 0.643 0.650 0.400 0.410 1.775 1.834 0.731 1.119 4.844 4.957 1.630 2.325

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[25] Table 3 – Model Performance One-Step Ahead Forecasts

The table shows the performance of the models in conducting one-step ahead forecasts. The ΔMSE is the mean squared errors of the benchmark model minus the mean squared error of the model. Since mean squared error is a loss function, a positive ΔMSE value indicates that the model outperforms the benchmark model. The p-values (P-Val) are calculated based on the two-sided Diebold-Mariano test. LTA represents the Long-Term Average of the volatility using a rolling window.

Model: Benchmark model:

∆ MSE P-Val ∆ MSE P-Val ∆ MSE P-Val ∆ MSE P-Val ∆ MSE P-Val ∆ MSE P-Val

ROYAL DUTCH SHELL 0.086 0.000 0.039 0.044 0.255 0.000 0.047 0.003 0.216 0.000 0.169 0.003

UNILEVER 0.175 0.000 0.118 0.000 0.052 0.000 0.057 0.000 -0.066 0.002 -0.123 0.000 HEINEKEN 0.100 0.000 0.110 0.000 0.086 0.000 -0.010 0.080 -0.024 0.034 -0.015 0.062 KONINKLIJKE PHILIPS 0.197 0.000 0.194 0.000 0.184 0.000 0.003 0.713 -0.010 0.609 -0.013 0.404 KON. AHOLD 0.070 0.141 0.110 0.081 0.034 0.322 -0.040 0.147 -0.076 0.121 -0.035 0.275 AKZO NOBEL 0.301 0.000 0.255 0.000 0.134 0.000 0.046 0.038 -0.121 0.000 -0.167 0.000 AEGON 0.567 0.000 0.746 0.000 0.159 0.000 -0.178 0.000 -0.586 0.000 -0.408 0.000

KON. BOSKALIS WEST 0.519 0.000 0.894 0.000 0.169 0.000 -0.375 0.000 -0.725 0.000 -0.350 0.000

SBM OFFSHORE 0.736 0.000 0.738 0.000 0.279 0.000 -0.001 0.979 -0.458 0.000 -0.457 0.000

KON. BAM GROEP 1.176 0.000 1.283 0.000 0.505 0.000 -0.107 0.000 -0.778 0.000 -0.671 0.000

CORBION NV 0.226 0.000 0.234 0.000 0.096 0.000 -0.008 0.697 -0.138 0.000 -0.130 0.002 WERELDHAVE 0.087 0.000 0.093 0.000 0.057 0.000 -0.006 0.123 -0.035 0.071 -0.030 0.087 KON. WESSANEN 0.223 0.000 0.228 0.000 0.106 0.000 -0.006 0.094 -0.123 0.000 -0.117 0.000 WOLTERS KLUWER 0.078 0.000 0.087 0.000 0.088 0.000 -0.009 0.266 0.000 0.924 0.010 0.409 RELX NV 0.064 0.000 0.065 0.000 0.076 0.000 -0.002 0.675 0.010 0.433 0.012 0.378 BINCKBANK 0.350 0.000 0.550 0.001 0.052 0.261 -0.200 0.014 -0.498 0.002 -0.297 0.001 ORDINA 0.182 0.335 0.195 0.304 0.209 0.005 -0.013 0.814 0.014 0.994 0.027 0.904 AALBERTS 0.118 0.000 0.195 0.000 0.401 0.000 -0.077 0.000 0.206 0.000 0.283 0.000 KON. DSM 0.178 0.000 0.150 0.000 0.119 0.000 0.028 0.000 -0.031 0.116 -0.059 0.000 RANDSTAD HOLDING 0.334 0.001 0.524 0.000 0.186 0.000 -0.191 0.000 -0.338 0.008 -0.148 0.120 ING GROEP 0.547 0.000 0.582 0.000 0.588 0.000 -0.035 0.451 0.005 0.972 0.041 0.726 FUGRO 2.177 0.000 1.348 0.000 0.524 0.033 0.829 0.089 -0.824 0.013 -1.653 0.004 KON. KPN 0.171 0.000 0.179 0.000 0.069 0.000 -0.007 0.236 -0.110 0.000 -0.102 0.000 ASML HOLDING 0.363 0.000 0.306 0.000 0.174 0.000 0.057 0.000 -0.133 0.000 -0.189 0.000 ASM INTERNATIONAL 0.222 0.000 0.280 0.000 0.249 0.000 -0.058 0.000 -0.031 0.433 0.027 0.214 KON. VOPAK 0.237 0.000 0.260 0.000 0.072 0.000 -0.023 0.034 -0.188 0.000 -0.165 0.000

Implied Volatility GJR-GARCH GARCH

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remain in the time series for up to a month. Moreover, both the ARCH-type models appear to have a higher mean value. This is especially clear from the later, more stable, months in our forecasts. During this period, the forecasts are almost consistently 1% higher than the implied volatility forecasts and the observed range based volatility. These two characteristics drastically increase the mean squared error.

From appendix 14 we can observe that in the case of Fugro the realized range based volatility in the last few months of our forecasts, was very close to the long-term average (LTA) of the volatility. Such that the mean squared errors for the long-term average are minimal. This causes the LTA to outperforms GARCH in 19 of 26 companies. Additionally, of the 7 cases where the GARCH performs better, only 2 are significant (Royal Dutch Shell and Aalberts).

5.4. Different Horizons and Windows

One can observe that as the k-horizon increases the volatility of volatility drops, this smoothens the distribution. In other words, large peaks in the time-series become smaller as the (k) horizon increases. This can be observed from appendix 11 which shows that the largest peak in the volatility of Fugro (March 2016) is around 17% for k=1, 30% for k=5, and 60% for k=20. Note that while we would expect the volatility to increase with the square root of time, the observed increase is actually smaller.

This is in line with Andersen, Bollerslev, Christoffersen, and Diebold (2006) who show that the term-structure of variance flattens as k increases. Meaning that for increasing values of k the

standardized daily variance moves to the long-term unconditional variance. This is visualized

in appendix 10 for the k-period forecasts of Ordina.

When we look at the h-step ahead forecast we find that for larger values of h the mean squared errors get larger. This is partly due to the fact that a shock to the realized volatility will not show up in our estimate until h-days later. This results in high squared errors as large spikes in volatility generally only last for a few days.

What is unexpected though is that the volatility forecasts in relatively calm periods are higher for larger values of h. As an example, let’s take a look at the forecast of Fugro in appendix 11. In the later months of our forecast (2017 specifically) we observe a relatively calm period, in which the forecasts of h=1 are close to the range based volatility of ~2%, yet for h=20 the forecasts are ~3% (while the range based volatility is of course unchanged). This result is also observable from the mean squared error decomposition in appendix 19, where the (GJR-) GARCH forecasts exhibit an increasing bias as the horizon (h) increases.

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window in table 2. Moreover, the sign of the difference changes per company, such that there is no decisive result on which window performs best. This is even the case for the financial institutions which showed large volatility spikes in 2009, for which we thus expected the windows to differ. As the differences among the recursive and rolling window are miniscule, no further significance tests will be conducted, as it is unlikely that these will grant us any additional information or insights.

5.5. Possible Explanations

As discussed earlier, volatility shocks persist in the forecast for a long time in the GARCH (and GJR-GARCH) model. This tremendously inflates the mean squared errors and at first glance one would think there is a (small) misspecification in the model. Yet, when looking at the historical volatility of Fugro a similar volatility spike can be observed in November 2014. Unlike the more recent one, this spike in volatility lasted for nearly a month. As such, one would expect future volatility spikes to persist in the data as well, which explains (part of) the slow decay in our forecasted series.

This is in line with the observed average beta of 0.806 and 0.886 for the GARCH and GJR-GARCH models of Fugro, respectively.6_{While all the time series are mean-reverting (since α} + β < 1), the large beta means a lot of weight is assigned to the lagged variance, which causes spikes to persist in the pattern for a relatively long period of time.

In addition to the slow decay, the bias-variance decomposition in table 4 shows that all GARCH models contain large upward biases, whereas the implied volatility model are almost unbiased. To such an extent that the largest ‘bias as % of total error’ in the implied volatility model —

which is clearly an outlier — is still lower than the lowest value of bias of the GARCH. Moreover, when comparing the variances in the bias-variance decomposition (table 4) we see that often the variance of the GARCH models are close to the variance of the IV models. Therefore, we can conclude that the main reason the implied volatility outperforms the ARCH-type models is that the IV model is a nearly unbiased estimator and the ARCH-ARCH-type has a large bias. We can actually observe from table 4 that for the companies where the GARCH performed relatively well (e.g. Royal Dutch Shell or Ordina) the ‘bias as % of total error’ is smaller. A possible explanation for the large bias might that there are structural breaks in the time-series. A structural break is when a time-series changes abruptly at a certain point in time, which can have a large effect on the parameters of the models. The possibility of a structural break (or breaks) is corroborated by the considerable difference in the range and standard deviation of the different samples: 2009 vs. 2010, and 2009 vs. the entire sample excluding 2009, which can be observed in appendix 4. Nonetheless, looking at the range of the estimated parameters of the

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[28] Table 4 – Bias/Variance Decomposition of Mean Squared Errors

Table shows the decomposition of the mean squared errors in the one-day ahead forecast. IV is the implied volatility model, LTA the long-term moving average of the historical range based volatility.

Var Bias^2 Var Bias^2 Var Bias^2 Var Bias^2 GARCH GJR IV LTA GARCH GJR IV LTA

ROYAL DUTCH SHELL 0.315 0.066 0.298 0.036 0.292 0.003 0.525 0.025 17.4% 10.8% 1.0% 4.6% + + +

-UNILEVER 0.370 0.111 0.327 0.097 0.303 0.002 0.353 0.005 23.0% 22.9% 0.8% 1.3% + + + + HEINEKEN 0.149 0.110 0.154 0.115 0.150 0.009 0.173 0.071 42.3% 42.7% 5.6% 29.0% + + + + KONINKLIJKE PHILIPS 0.221 0.191 0.227 0.183 0.208 0.007 0.247 0.153 46.4% 44.6% 3.2% 38.2% + + + + KON. AHOLD 0.357 0.053 0.390 0.060 0.340 0.000 0.374 0.000 12.8% 13.4% 0.0% 0.0% + + + -AKZO NOBEL 0.347 0.244 0.318 0.227 0.284 0.006 0.347 0.078 41.3% 41.6% 2.2% 18.4% + + + + AEGON 0.915 0.426 1.025 0.494 0.773 0.000 0.890 0.043 31.8% 32.5% 0.1% 4.6% + + - +

KON. BOSKALIS WEST 0.709 0.312 0.912 0.484 0.494 0.008 0.566 0.106 30.6% 34.7% 1.6% 15.7% + + + +

SBM OFFSHORE 0.684 0.594 0.680 0.598 0.540 0.000 0.786 0.034 46.5% 46.8% 0.0% 4.1% + + - +

KON. BAM GROEP 0.607 1.256 0.622 1.347 0.586 0.100 0.728 0.464 67.4% 68.4% 14.6% 38.9% + + + +

CORBION NV 0.769 0.176 0.806 0.146 0.714 0.005 0.764 0.051 18.6% 15.4% 0.7% 6.3% + + + + WERELDHAVE 0.153 0.084 0.157 0.085 0.141 0.008 0.172 0.035 35.6% 35.2% 5.5% 16.8% + + + + KON. WESSANEN 0.580 0.197 0.586 0.197 0.552 0.003 0.587 0.073 25.4% 25.2% 0.5% 11.1% + + - + WOLTERS KLUWER 0.180 0.069 0.189 0.069 0.167 0.003 0.219 0.039 27.6% 26.7% 1.9% 15.2% + + + + RELX NV 0.198 0.056 0.195 0.061 0.186 0.004 0.235 0.030 21.9% 23.7% 1.9% 11.3% + + + + BINCKBANK 0.685 0.254 0.845 0.294 0.547 0.042 0.633 0.008 27.1% 25.8% 7.1% 1.3% + + + + ORDINA 1.855 0.333 1.899 0.302 2.005 0.001 2.181 0.034 15.2% 13.7% 0.1% 1.5% + + + + AALBERTS 0.278 0.116 0.302 0.169 0.255 0.022 0.301 0.376 29.4% 35.9% 7.8% 55.5% + + + + KON. DSM 0.267 0.151 0.259 0.131 0.237 0.003 0.303 0.056 36.1% 33.6% 1.2% 15.7% + + + + RANDSTAD HOLDING 0.525 0.279 0.624 0.370 0.470 0.000 0.565 0.091 34.7% 37.2% 0.0% 13.8% + + + + ING GROEP 0.822 0.334 0.879 0.312 0.608 0.001 0.755 0.441 28.9% 26.2% 0.2% 36.9% + + - + FUGRO 2.611 0.908 1.837 0.853 1.307 0.035 1.821 0.045 25.8% 31.7% 2.6% 2.4% + + + -KON. KPN 0.411 0.147 0.417 0.149 0.385 0.001 0.434 0.021 26.3% 26.3% 0.2% 4.6% + + + + ASML HOLDING 0.317 0.332 0.319 0.273 0.285 0.002 0.347 0.113 51.1% 46.1% 0.6% 24.6% + + + + ASM INTERNATIONAL 0.502 0.186 0.542 0.203 0.465 0.001 0.522 0.193 27.1% 27.3% 0.2% 27.0% + + - + KON. VOPAK 0.378 0.208 0.393 0.216 0.345 0.004 0.407 0.013 35.5% 35.5% 1.2% 3.1% + + + +

One-Step Ahead Forecasts

Sign of bias Bias as % of Total Error

LTA IV

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[29]

GARCH models (appendix 16), there don’t seem to be large differences in the parameter estimation, which we would expect if 2009 was in fact different than the rest of the sample.7 A possible structural break could result in a large bias, which in turn causes the GARCH to perform poorly out of sample. This is somewhat in line with Figlewski (1997) who argued that although GARCH performs well in explaining in-sample volatility, it explains little of the volatility in forecasts of ex-post squared returns. On the contrary, Andersen and Bollerslev (1998) argue that the failure of the GARCH model to forecast future volatility is not due the GARCH itself, but rather due to failing to specify a correct realized volatility measure. This brings us to the last possible explanation for the poor performance of the GARCH models: the suitability of the range based volatility.

Since volatility is a unobserved variable we used range based volatility as a proxy. While Alizadeh, Brandt, and Diebold (2002) found that generally range based volatility is a suitable proxy for volatility if intraday data is not available. However, Patten (2011) found that while the range based volatility leads to less distortion (compared to daily close-to-close volatility), the degree of distortion was often still large.

Nevertheless, when we compare the volatility forecasts obtained from the GARCH with the close-to-close absolute returns, we find that its performance is even worse. 8 Indicating that while the range based volatility might not be the perfect estimate, it certainly is more suitable than the commonly used close-to-close volatility.

Figure 5 - One-Day Ahead Forecasts AEX Index

Forecasts for the AEX index using the GARCH, GJR-GARCH and IV models. Range denotes the range based volatility. The AEX price and option data were collected through DataStream in similar fashion as the main sample (see section 3).

7_{A few clear exceptions apply, see e.g. Ordina, Kon. Ahold, or Binckbank.}

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[30] 5.6. AEX Index

The data shows that the return series of individual companies tends to contain large sudden spikes of volatility. Moreover, we found that a reason the ARCH-type models perform relatively poorly is that these sudden shocks persist in the ARCH models’ predictions for a relatively long time. It would therefore be interesting to see whether this is also the case for the aggregate of these series – the AEX index. As it is likely that by taking the aggregate of the individual series the extreme events are partly ‘averaged out’ and will thus lead to less immense shocks and lower volatility of volatility.

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[31]

6. Conclusion

This paper conducts a comparative study in which three volatility forecasting models are tested. These are (1) the GARCH model; (2) the GJR-GARCH model; and (3) the option implied volatility model.

Based on the theory and previous literature two hypotheses were formulated. Null hypothesis A states that GJR-GARCH does not outperform the GARCH, it’s corresponding alternative hypothesis is that the GJR-GARCH does outperform the GARCH. Null hypothesis B states that the option implied volatility is not a better estimator of volatility than the ARCH-type model, the alternative hypothesis is that it is. These hypotheses were tested by conducting out-of-sample forecasts for 26 Dutch listed stocks.

From the out-of-sample forecast can be concluded that the implied volatility model is the preferred method for forecasting volatility for every company in the sample. Moreover, no compelling evidence was found that the GJR-GARCH outperforms the GARCH. Indicating that the asymmetry term in the GJR-GARCH does not convey any additional information. More formally, we can state that we failed to reject null hypothesis A and do reject null hypothesis B.

Subsequently, the paper addresses some possible explanations for these results, these are: 1) The persistence of shocks in the ARCH-type models; 2) the fact that ARCH-type models are biased estimators and the IV nearly unbiased; 3) the possibility of structural breaks; and 4) the suitability of the range based volatility as a proxy for realized volatility.

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[35]

Appendices

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[36] Appendix 1 - Black-Scholes Formula

Forecasting Volatility of Dutch Stocks: A Comparative Study on the Forecasting Performance of Option Implied Volatility and ARCH-type Models.