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Master thesis, 15 hp Master’s in economics, 15 hp

Spring term 2020

Can we predict future volatility on the OMXS 30?

A quantative study on historical and implied volatility

Martin Hallberg

Martin Hallberg Umeå University Master thesis 1, 15 hp

Abstract

When making investment decisions risk is a highly important aspect to account for. Many

studies have investigated how to measure risk and forecast it for an investment decision. This

study takes a closer look at what forecast method is best on the Swedish index OMX Stockholm

30. During the period from January 2016 to December 2018. The models examined are

GARCH, EGARCH and Black-Scholes (implied volatility). The result indicates that EGARCH

is best at forecasting proxies for the index volatility. All models follow the realized volatility

proxies fairly well, but implied volatility constantly overestimates the volatility. This is

consistent with previous research.


Innehållsförteckning

1 Introduction ................................................................................................................................................. 1 1.1 Background ................................................................................................................................................... 2 1.2 Problem definition ........................................................................................................................................ 2 1.3The object of this study .................................................................................................................................. 3 1.4 Limitations .................................................................................................................................................... 3 1.5 Method description ....................................................................................................................................... 3

2 Literature review ......................................................................................................................................... 4 2.1 What is an option? ........................................................................................................................................ 4 2.2 What is volatility ........................................................................................................................................... 5 2.3 Forecasting volatility and previous studies ................................................................................................... 6

3. Empirical method and Data ....................................................................................................................... 10 3.1 Data 10

3.1.1 Explanation of variables ...................................................................................................................... 10 3.1.2 Measurements .................................................................................................................................... 10 3.1.3 Table of data: ...................................................................................................................................... 11

3.2 Box Ljung test ............................................................................................................................................. 11 3.3 ARCH-model (Autoregressive conditional heteroskedasticity) .................................................................... 12 3.4 GARCH-Model. (Generalized Autoregressive conditional heteroskedasticity) ............................................ 12 3.5 EGARCH (Generalized Autoregressive conditional heteroskedasticity) ...................................................... 14 3.6 Black-Scholes model ................................................................................................................................... 15 3.7 Implied volatility ......................................................................................................................................... 16 3.8 Realized volatility ........................................................................................................................................ 17 3.9 Forecast Garch and Egarch. ........................................................................................................................ 18 3.10 Evaluation of forecasts ............................................................................................................................. 19

4. Results ...................................................................................................................................................... 20 4.1 Box-Ljung test ............................................................................................................................................. 20 4.2 Forecast evaluation .................................................................................................................................... 20 4.4 Summary of estimated variables ................................................................................................................ 22 4.5 Correlation of estimated variables. ............................................................................................................ 23

5 Discussion .................................................................................................................................................. 24

6. Conclusion ................................................................................................................................................ 27

7 References ................................................................................................................................................. 28 7.1 Books .......................................................................................................................................................... 28 7.2 Studies ........................................................................................................................................................ 28


Appendix 8 ................................................................................................................................................... 30 Appendix 8.1 Autocorrelation function(ACF) and Partial Autocorrelation Function(PACF) .............................. 30 8.2 Distribution of data ..................................................................................................................................... 31 8.3 GARCH and EGARCH-models ...................................................................................................................... 31


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1 Introduction

This section will include our purpose and problem definition to give the reader a clearer understanding of what implications that founded this study.

To get the most efficient allocation of resources in society we need to have an efficient financial

market. The efficiency of resource allocation is highly dependent on accurate risk measures.

When evaluating risk of the stock market we often use the standard deviation of returns. The

question is if we forecast this measure any better?

Today multiple models that are used to price options. Most commonly used are the Black-

Scholes-model and variants of this model. In the Black-Scholes, we get all parameters except

for volatility quoted from the market. After an option has been entered into a market it will

find an approximative equilibrium price. From this price, we can extract information on what

the market participants think about future market volatility. This information is important

because this measure is the standard deviation of returns. With this information, we get a

measure of the expectations of a risk measure there is in the market.


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1.1 Background

Pricing derivatives are really important for market participants that want to invest in the market

by evaluating the risk. A common way to use derivatives is to hedge, for example, if I own a

stock and don’t want to be exposed to the stock’s price fluctuations. I can then buy a short

position, theoretically selling that stock, then buy it back, theoretically the future moves in the

opposite direction of the stock price. Then the trader is not exposed to the price movements but

still owns the stock. Another derivative is options, that give the buyer the right, but not the

obligation, to buy an asset at a certain price. This is very common in the commodity market and

later years become more common in the stock markets. This type of financial product is

important to allocate the recourses of society to their most effective location.

1.2 Problem definition

The stock market is complex and developing a measure of market expected volatility can prove

to be a good forecast for the future. This to be able to improve forecasts and decision-making.

Today there are good models to predict volatility from historical data. The implied volatility is

mostly based on market expectations, supply, demand and market arbitrage possibility on the

options. This will give me a forecast from expectations to compare with historical data models.

The study of forecasting volatility is highly investigated end researched. There are many

different kinds of models for this purpose. Models that are usual when predicting volatility are

Garch-models (Generalized autoregressive conditional heteroskedasticity). This type of

modeling uses historical data to predict what will happen in the future. There are many studies

with the scope comparing these models. This study will investigate, compare and analyze the

forecastability of volatility on the Swedish OMX Stockholm 30 index.


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1.3 The object of this study

The object of this study is to forecast volatility from two datasets stemming from expectations

(implied volatility) and historical volatility (Garch-modeling). This to answer the following

questions:

- Can we accurately forecast future volatility on the OMXS 30 index?

- What method provides the best forecasts for future volatility on the OMXS 30 index?

1.4 Limitations In this study, I will analyze the predictive power of returns and implied volatility on future

volatility. The data collection is limited since most of the large datasets that would be perfect

comes with a big cost. Therefore, I will use the data I can find on Thomas Reuther Datastream

that I can get free from campus. The only time-period of the data that is available is from 2016

to the current day. Because of the structure and length of the option data that is available the

data management will take up a very large portion of the time. Since most of my time will go

towards data management in turn this can affect the model difficulty.

1.5 Method description To answer these research questions, I will do a quantitative study with multiple forecasting

models. Because of the time-series data, I will do time-series modeling with two Garch type

models. For the implied volatility I will use the Black-Sholes option pricing model to extract

the expectation, forecast, of volatility. To compare these forecast I will use four different

statistical error measures to see what forests deviate from the actual realized volatility on the

OMXS 30 index.


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2 Literature review

In this section, I will describe and define options and volatility to allow the reader an opportunity to understand my research better. Further, I will analyze previous studies and theses to get a better understanding of what knowledge there is in this field.

2.1 What is an option?

A derivative is defined as a financial instrument whose value depends on, is derived from,

another financial instrument. The trading of derivatives has increased dramatically in later

years. One type of derivative is named options, a contract between two parties where one is a

writer and one is a holder. There are two types of basic options a call, right to buy the underlying

asset, and a put, right to sell the underlying asset at a given strike price. If an investor buys a

call option and at the expiration date the underlying asset price is higher. Then the investor has

the right, but not the obligation, to exercise the option to buy the asset cheaper and the writer

must sell to that price. These types of basic options are a way to transfer risk, like insurance.

Another use of options is to hedge against price fluctuations and netting out positions (Cecchetti

and Schoenholtz 2015, 220-225).

There are terms often used when talking about options. The strike price is the price that the

holder can buy the underlying asset to at expiration. Time to maturity is days left to the

expiration of the option. At the money (ATM) is when the underlying asset is at the strike price.

Out of the money (OTM) when calls (puts) strike price is below (above) the underlying asset

value. In the money (ITM) when calls (puts) strike price is above (below) the underlying asset

value. There are multiple types of options, the one used in this article is a European type option.

These cannot be exercised before the expiration date. (Hull 2012, pp. 7-9)


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2.2 What is volatility

Volatility forecasting is a really important subject and has been highly studied because of its

importance in assets valuation and risk management. Volatility is are the most important

parameter when valuation market options since it often will have the largest price effect.

Volatility is often defined as the return distribution and the most common measure is the

standard deviation, definition in equation 1 below. Since most distributions measure variability

with the standard deviation this is the most common measure for volatility. When forecasting

volatility, the data is suggested to use should at least be as long as the period that will be

forecasted. The variance is a more unbiased estimate, but if we model on 𝜎" this will become

biased because of Jensen’s inequality, defined in equation 2 (Poon and Granger 2001)

𝜎"! = "#$"

∑ (𝑟% − �̅�)!&%$" Equation (1)

Were n being the number of observations, r is the return and 𝜎 is the variance.

Jensen’s inequality states that:

𝐸+𝑓(𝑥). > 𝑓(𝐸(𝑥)) Equation (2)

f is a non-linear function of x.

Volatility is not directly observed, only estimated, but it has many important applications in

finance. Since we can observe asset prices in the market these are used to measure and predict

volatility with different measures. Volatility often has clusters of volatility with low and high

volatility. Since volatility evolves, over time, large quick fluctuations in volatility are rare.

Volatility does not have an endless range and therefore is often stationary, i.e. not time-

dependent. Large price drops are said to have a greater effect on volatility than an equally large

price increase, this is refed to as leverage effect (Tsay p.177).


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2.3 Forecasting volatility and previous studies

In the research by Kambouroudis, McMillan and Tsakou (2016) who analyzed forecasting stock

return volatility. Looking at the predictive power of implied volatility compared to different

GARCH-models on stock market returns. They did this on data from S&P Composite 500, Dow

Jones Industrial Average (DJIA) and Nasdaq 100 and volatility indices of these indices. The

findings of this study point towards both the implied volatility and realized volatility models

have a large predictive power of future volatility. Implied volatility (IV) forecasts are found to

be significant. IV models account for the contemporaneous asymmetric effects, leverage

effects, and forecasts of this type outperformed the random walk model. When making a model

of both IV, RV and Garch-models forecast this performed the best.

McMillan and Evans (2007) did a study in forecasting volatility with five different Garch-

models and similar forecasting models they accomplished this in 33 countries, Sweden among

them. During the period from 1994 to 2005 on the country’s main indices, in Sweden OMXS

30. Different iterations of the Garch-models were often found to be optimal, in Sweden the

Egarch-model performed the best.

Antonucci (2008) investigated the best predictor of volatility by comparing implied volatility

and different Garch-Models. By looking at WTI options (Crude Oil options) on the NYMEX

(New York Mercantile Exchange). He used WTI options in his study because of the large

turnover of these derivatives. The models used that was GARCH, ARCH, EGARCH,

CGARCH and TGARCH. He also used different distributions in estimations of their models

to compare the difference. After predicting and evaluating the predictions with statistical

regression-based criteria. They found no leverage effect in the modeling, GARCH type models

were found to forecast better than implied volatility models. The GARCH-models with

generalized error distribution performed the best when evaluated with mean square error (MSE)

and mean absolute error (MAE). Looking at the error distribution only a marginal increase in

forecasting performance was observed.


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In 2004 Verhoeven and McAleer investigated the effects of fat tails and asymmetry in financial

volatility modeling. By estimating GARCH-models on indices in the USA, Australia, Kuala

Lumpur with data from 1990 to 2000. They found that there are benefits to estimating

conditional mean-variance models using conditional non-time-varying asymmetric leptokurtic

distributions. If the time-series exhibits large variance and have a higher degree of non-

normality. Continuously find that constant skewness of the unconditional returns not improve

when modeling with time-varying asymmetries.

Phoon and Granger (2001) did a sizable literature revive of 72 studies and papers that evaluate

and analyzed time series modeling and implied volatility. This to evaluate if volatility is

forecastable and what methods forecast the best. They discuss the importance of turnover and

why the equilibrium price is not always quoted in the market. The reasoning is that because of

frictions in the market the price will not always reach the equilibrium. Examples of these types

of frictions are transaction costs, non-continuous trading, block trades, bid-ask spread and tick

size. Market frictions can reduce the market efficiency in pricing and therefore the implied

volatility measure. These frictions are found to have a large price effect on options that are deep

ITM or deep OTM. In other words, small fluctuations of the underlying asset can result in large

effects of volatility and therefore price. There are many iterations of the Black-Scholes model

and can suffer from two types of errors: underlying asset distribution mismatch and clientele

effect in options trading. They found that because of different data, time periods and different

evaluation strategy the comparison of studies was difficult. A common approach found to be

used was to compare different error measures of the forecasts.

Poon and Granger conclude that non-linear GARCH-models often outperformed similar

simpler models. They found that most study’s analyzed stated that implied volatility provided

large information about future volatility and often outperformed GARCH-models. This

grounded in theory from that the implied volatility contains information that the historical

volatility doesn’t. However, that implied volatility is a more biased estimator. The theoretical

expectations are that implied volatility should perform better than since that measure takes

future information onto account. While historical data modeling will only take historical

information into account.


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The bias discussed above was researched by Neely (2009). Much of the bias was said to come

from not using priced volatility and that high-frequency data can’t mitigate this bias. The option

price is dependent not only on price fluctuations of the underlying asset but from the volatility

fluctuations. If an investor uses an option to hedge against price fluctuations in the underlying

asset the investor will still be exposed to the volatility risk. This volatility risk is said to be

systematic and the investor must be compensated for the risk exposure. Thus, if the market

price from options has priced this volatility risk the implied volatility measure is more likely to

overestimate the volatility. Even when using econometric modeling to account for this bias the

article failed to explain a significant part of the many different biases. He concludes by saying

that the implied volatility measure is biased but efficient in estimating volatility. He believes

that statistical metrics are inappropriate measures for the information content of implied

volatility.

In an article by Bakshi, Cho and Chen (2000) they analyze the movement of call and put options

compared to the underlying asset. Using high-frequency data from the S&P 500 from March 1st

1994 to August 31st 1994 and using one-dimensional diffusion option models. Findings point

towards that majority of assumptions from the Black-Sholes model still hold under a general

one-dimensional diffusion setting. Theoretically, the price movements of the underlying asset

and the option, dependent on the model, should exhibit high if not perfectly correlated. They

find that the prediction of these models used often violates this theoretical statement.

Continuously, the microstructure effects (like bid-ask spread, tick size restriction, and so on)

can be a contributing factor to disproportional movements and no effects on the option price

when there are price movements in the underlying asset.

Christensen and Parbhala, (1997) investigated if the implied volatility were inefficient and

biased by using data from options on the S&P 100 index. By comparing implied volatility to

realized volatility proxy they find that implied volatility on a monthly basis was effective.

Findings indicate that the measurement to be good and not biased. They discuss how abnormal

events such as crashes can affect the credibility of implied volatility forecasts.


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Green and Figlewski (1999) analyze the market risk and model risk for financial institutions

writing options. This by studying the Black-Scholes model and cash-flow matching techniques.

Using historical data modeling to forecast volatility’s, to subsequently evaluate the forecasts

using root means square error (RMSE). When looking at the risk exposure they find that model

risk is quite large. Findings point towards that the cash-flow matching strategy is sound if it can

be done precisely because it removes the model risk entirely. But since the market often wants

to hold long positions in stock the institution must then take large short positions on the balance.

Dealers are then forced to hold exposed, unmatched, option positions. They find modeling with

Black-Sholes to be a reasonable strategy but highlight the impotence of correlation in assets to

offset losses by delta hedging. Delta hedging is when you hedge against the directional risk of

an asset by taking a position in an asset that moves in the opposite direction. Continuously

stating that the delta hedging from the Black-Scholes model removes a sizable part of the

market risk but still faced with there is a model risk. Institutions and market participants will

not offer the price at equilibrium to the market since this will be the expected break even.

Therefore, mark up the prices of the asset to offset losses from other risk factors.

Hansen and Lunde (2005) investigate if the GARCH(1,1)-model is ever outperformed on

exchange rates and IBM stock returns. They do this by testing against 330 similar or more

sophisticated models. This to forecast variance out-of-sample and compare it with a realized

volatility proxy. The findings state that the Garch(1,1) model is inferior when modeling on IBM

returns. Findings imply that GARCH-models that account for leverage effect is superior to

simpler GARCH-models.


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3. Empirical method and Data

In this section, I will present and explain the empirical models, the data collections and treatment prosses. Further, I will discuss the problems of the methods and ways to reduce the problems of weaknesses.

3.1 Data

For this thesis, I have collected data available from Thomas Reuters Datastream. The dataset

collected contains OMXS 30 (OMX Stockholm 30) price levels from 2010- 2018. I also

collected intraday high price and intraday low price during the same period. The OMXS 30

index is a Swedish stock market index comprising of the 30 largest turnover firms. Because of

limitations of option data available and time to manage the data I have collected individual

prices, expiration date and strike price of options on the OMXS 30 index from 4th January 2016

to 28th December 2018. The options data collected all options have time to maturity of 1 week

or less, all options stand at the money or as close to at the money as possible. Mitigate the

friction of low turnover by having options close or ATM. The dataset contains Stockholm

Interbank Offered Rate with a 1-week interstate that is matched to each option.

3.1.1 Explanation of variables

RET- Return of the OMXS 30. From Close to close return.

OMXH – OMXS 30 intraday high price.

OMXL – OMXS 30 intraday low price.

OMX – OMXS 30 price at the close.

R – Stockholm interbank offered 1week rate.

3.1.2 Measurements RV – Realized volatility. The close to close Absolut return of the underlying asset.

IDRV- intraday realized volatility. The log of the intraday high price devised by the intraday

low price.

IV – Implied volatility. Calculated from the Black-Scholes model.

GARCH – Forecasted values from the standard GARCH- model.

EGARCH – Forecasted values from the EGARCH-model.


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3.1.3 Table of data: This is the data collected from the OMXS 30 index. Table 1. Descriptive statistics

Variable N.obs Mean Sd. Dev. Min Max

RET 2347 7,257E-05 0,00495847 -0,038219 0,02707881

OMXH 2347 1325,216 229,9089 873,66 1720,02

OMXL 2347 1308,478 229,9576 833,03 1699,48

OMXS 2347 1317,787 229,0961 862,17 1719,93

R 2347 0,503043 1,00767 -0,778 2,45

IDRV 751 0.082881 0.043364 0.021919 0.35010

RV 751 0.050612 0.044931 0.0002280 0.30670

We can see that the mean return is approximately zero. The intraday realized volatility and the

realized volatility mean are very different but with large standard deviations. The number of

observations of RET, OMXH, OMXL, OMXS and R is longer from 2010 to 2018 to fit the

Garch model. The IDRV and RV are during the period 2016 to 2018.

3.2 Box Ljung test

To evaluate the data for the modeling, I must investigate if the data is independently identically

distributed (iid). To do this I will use the Box Ljung test.

The test statistic is of the Box-Ljug test is:

𝑄 = 𝑛(𝑛 + 2)∑ '(!"

(#$*),*-" Equation (3)

The test hypothesizes that the data is independently distributed. In other words that the data do

not have constant variance. One of the assumptions of the Garch-models is that the data should

be iid and display autocorrelation. The alternative hypothesis is that the data is not

independently distributed. Ljung and Box (1978) ‘


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3.3 ARCH-model (Autoregressive conditional heteroskedasticity)

ARCH was developed by Engle (1982). The fundamental idea of the model is that the data is

stationary, i.e. not time-dependent, but dependent on volatility in the errors. This means that

high variance is followed by high variance in the error of prediction. This does not insecurely

say that large variance is followed by large variance but that the probability is larger for this to

be true. The ARCH(p) model can be dependent on multiple lagged values. The ARCH-model

formula is defended by equation 4.

s%! = a. + a"𝑎%$"! +⋯+a#𝑎%$#! 𝑤𝑒𝑟𝑒𝑎% = s%e% Equation (4)

Where e is an independent and identically distributed (iid) random variable with a mean of zero

and a variance of 1. While using the ARCH-model it is often assumed that e is distributed after

the student t distribution or a generalized error distribution. Return data is often exhibiting

kurtosis, fat tails in the distribution, therefor we often model the ARCH-model on the sample

data distribution. There are some advantages to the ARCH-model, the model can produce

volatility clusters, when the shocks in time t of the model have heavy tails. The negative aspect

of this model is that large positive variations in returns and large negative variations are known

not to have the same effect on future volatility. (Tsay 2015, p.184-197)

3.4 GARCH-Model. (Generalized Autoregressive conditional heteroskedasticity) The ARCH-model is simple but unfortunately often needed many parameters to describe the

variance in asset returns. To still keep the model simple but more effective Bollerslev (1986)

developed a Generalized ARCH-model (GARCH-model). If we take a log-returns and let 𝑎% =

𝑟% − 𝜇% to be the innovation at time t. Thereby 𝑎% follows a GARCH(m,s) defined by equation

5.

𝑎% = 𝜎%𝜀%𝑤ℎ𝑒𝑛𝜎%! = 𝛼. +∑ 𝛼/𝑎%$/!#/-" +∑ 𝛽0𝜎%$0!0

1-" Equation (5)


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This in turn will make one lag contain more information on previous lags. Where e is iid with

a mean of 0 and variance of 1. The constraint 𝛼/ +𝛽/ < 1 implies that the unconditional

variance is finite, but the conditional variance 𝜎%! evolves, over time. In other words, we can

see that the conditional variance is dependent on the long-run average variance in the 𝛼., the

previous asset shock through ∑ 𝛼/𝑎%$/!#/-" (from the ARCH-model) and the previous variance

through the term ∑ 𝛽0𝜎%$0!01-" . As in the ARCH-model e is assumed to be a student t distribution,

generalized error distribution or a fat-tailed generalized error distribution. The a0 is the ARCH

parameter and b0 is the GARCH parameter. The estimation of these parameters will result in

𝛼. > 0, 𝛼/ ≥ 0𝑎𝑛𝑑𝛽0 ≥ 0. A commonly used model is GARCH(1,1) since if we 𝑎%$" from

an estimation that depends on the lag 𝑎%$! and so on. (Tsay 2015, p.199-202)

The models that are used will be evaluated in conjunction with the information criterion. I will

also evaluate my models with autocorrelation function and partial-autocorrelation function,

found in appendix 8.1. The distribution of data for the Garch-model will be evaluated and fitted,

presented in appendix 8.2. Hanson and Lunde (2005) argued that the Garch(1,1) model is

inferior and I will, therefore, do a Egarch model to account for leverage effect.

The AIC (Akaike information criterion) is commonly used when evaluating lags and models.

The AIC tells us what errors the model fit exhibits. The AIC will help me choose the model and

is defined below:

𝐴𝐼𝐶(𝑝) = ln J223(4)5

K + (𝑝 + 1) J!5K Equation (6)

P is the number of coefficients; T is the number of observations and SSR stands for the sum of

squares residuals. Since of the expression we want to minimize the AIC. (Stock and Watson

2015). Since my data do not exhibit a high degree of non-normality, the data follow a student t

distribution best, presented in appendix 8.2. The data display kurtosis, fat tails, but according

to Verhoeven and McAleer (2007) this will not significantly influence my forecasts.


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3.5 EGARCH (Generalized Autoregressive conditional heteroskedasticity)

The EGRCH-model was introduced by Nelson (1991). The conditional variance equation of

the EGARCH-model is defined in equation 7:

ln(𝜎%!) = 𝜛 + 𝑔(𝑧%$") + 𝛽 ln(𝜎%$"! ) Equation (7)

Were the asymmetric response function being Equation 8:

𝑔(𝑧%) = 𝜆𝑧% + 𝜑 Q|𝑧%| − SJ!6KT Equation (8)

Z is the standard expected return. If the variables 𝜑 > 0, 𝜆 < 0. This variable will in turn induce

larger (smaller) volatility response of negative (positive) shocks in returns, this is called

leverage effect. This is something that has been observed to be true in the stock market (Hansen

and Lunde 2005). Many GARCH-models are dependent on non-negative constraints on

parameter since negative parameters can result in negative variance. This constraint is

automatically addressed in the EGARCH-model since the model is in logarithmic terms. The

EGARCH-model has been found to often fit financial data better than GARCH-models. The

logarithmic functional form is often better than standard GARCH-models even when ignoring

the leverage effect. (Alexander 2001, pp. 79-81)


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3.6 Black-Scholes model

There is another way to derive the market volatility from the option pricing formula Black-

Scholes. Developed from the Black-Scholes formula by Murton (1973) continuously refed to

as the Black-Scholes formula. This formula and variants of it have been seen as the industry

standard when calculating option prices. The original formula has a few underlying

assumptions:

1. The stock price follows a process: dS = 𝜇*𝑆*𝑑𝑡+𝜎*𝑆*𝑑𝑧.Alsocalledageometric

brownmotion.

2. Shortsellingofsecuritiesispermitted

3. Notransactioncostsortaxesandallsecuritiesareperfectlydivisible

4. Nodividends

5. Norisklessarbitrageopportunities

6. Securitytradingiscontinuous

7. Therisk-freeinterestrate,r,isconstantandthesameforallmaturities.

The Black-Scholes model is used to price individual call and put options this model builds on

the volatility of the stock.

𝐶. = 𝑆.𝑁(𝑑1) − 𝐾𝑒$75𝑁(𝑑2) Equation (9)

𝑃8 = 𝐾𝑒$75𝑁(−𝑑2) − 𝑆.𝑁(−𝑑1) Equation (10)

𝑑1 =9:;#$! <=;7=

%"∗?

"<5

?√5 Equation (11)

𝑑2 = 𝑑1 − 𝜎√𝑇 Equation (12)

C is the call option price, P is the put option price, S is asset price, K is the strike price, r is the

rate, T is time to maturity and 𝜎is the variance. N(x) is the cumulative probability standardized

normal distribution with a mean of 0 and the standard deviation equal to 1. The N(d2)

distribution represents the probability that the option will be exercised in a risk-neutral world.

This equation is independent of risk preferences since it is not in the equation. The Black-

Scholes formula builds upon risk-neutral preferences. (J. C. Hull, pp. 299)


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3.7 Implied volatility Implied volatility can be calculated from the Black-Scholes, equation 9 above. Since price, rate,

strike price and rate can be observed and can be found on the stock market we can theoretically

solve this equation for volatility. By deriving the Black-Scholes formula with respect to sigma

and useing Newton-Rapson method to solve for volatility. In this thesis, I found the simplest

way was to do an iterative search. In other words, we use an algorithm to guess repeatedly until

the equation is solved. With this solution, we can see what the market participants think of

future volatility, until the option expiration date. If the options market is efficient all

information should be included in implied volatility such as World leader meetings, market

information, etc. Therefore, containing information that can’t be extracted from future historical

data. The implied volatility is often quoted by traders rather than the price because of its

importance (J. C. Hull, 2012. Pp 318-319) (Poon and Granger 2001).

One problem with the estimation of implied volatility from the Black-Scholes formula is that

it’s not linear, this will result in a biased estimator of volatility. If we use short term options

and use options that are close to or ATM, we can minimize this bias by reducing the change in

“volatility smiles”. (Bodie and Merton 1995) (Poon and Granger 2001)


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3.8 Realized volatility

To benchmark my forecasts, I will compare the forecasts against two realized volatility proxies.

There are many measurements of realized volatility. To find an accurate proxy for realized

volatility I will use simpler formulas to reduce noise from the measurements. Firstly, I will use

the RV-measurement, i.e. absolute return in yearly terms defined in equation 13. This

measurement is very simple but can miss volatility if there is large volatility during a day and

as long as the OMXS30 close at the same price level it will not show this measure. To account

for this problem, I will use another measure as a proxy for volatility. Intraday volatility

presented in equation 14. This is the measure that is mostly used in similar previous research

(Poon and Granger 2001). This proxy is not perfect because the measurement can miss large

price swings between days and bid ask-bounces (Patton 2010). The daily realized volatility

measurements are defined in daily terms and to be comparable with implied volatility therefore

I must multiply by the square root of 252 (252 is the yearly average trading days during my

period).

𝑅𝑉 = √252 ∗ ∑ ln b2&,(2&,)c Equation (13)

𝑆%,, is the intraday high price and 𝑆%,B is the intraday low price.

𝐼𝐷𝑅𝑉 = √252 ∗ ln( 2&2&*%

) Equation (14)

𝑆% is the closing price today and 𝑆%$"is the closing price yesterday.


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3.9 Forecast Garch and Egarch.

Using the Garch models to forecast one day will get estimations for my volatility. When

estimating the GARCH model I will get the model estimation from equation 15.

𝜎"%="! =𝛼". +𝛼""𝑎%! + 𝛽e"𝜎%! Equation (15)

When estimating the EGARCH model the estimation is defined by equation 16.

𝜎%="!f= exp(𝜛j) exp+𝑔"(𝑧%).𝜎%!DE Equation (16)

The implied volatility is already a prediction of the future because it is solved from the Black-

Scholes option formula. IV lets us know what the market thinks about future volatility until the

expiration of the option. The forecasted volatility will den be until the day of the expiration.


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3.10 Evaluation of forecasts To evaluate the forecasted values, I will use statistical error measurements to see how much the

forecast deviates from the volatility proxies. Common measurements used are mean square

error (MSE), mean absolute error (MAE), root mean square error (RMSE) and mean absolute

percent error (MAPE). Some of these measurements have multiple definitions were the

observations and mean are squared. Since it is hard to evaluate common forecasts´ distribution

and I have returns that are known to exhibit kurtosis and fat-tails. Therefore, I will only use

standard measurements without squared variables. This because the confidence interval of the

error distribution can be wide when measuring from variances. Squaring these variances will

only exacerbate the problem. (Poon and Granger 2011) (Vee, Gonpot and Sookia, 2011) The

definitions of the error statistics iv used are:

𝑀𝑆𝐸 = (𝑅𝑉/ − 𝜎"%)! Equation (17)

𝑀𝐴𝐸 = "#∑ |𝑅𝑉/ −#/-" 𝜎"%| Equation (18)

𝑅𝑀𝑆𝐸 = S"#∑ (𝑅𝑉/! − 𝜎"%!)!#%-" Equation (19)

𝑀𝐴𝑃𝐸 = "#∗ 100 ∗ ∑ |3G+$?(&|

3G+#%-" Equation (20)

𝑅𝑉/ is the realized volatility proxy and 𝜎"% is forecasted from GARCH, EGARCH and IV.


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4. Results

In this section, I will introduce the test results, findings in the thesis and summarize the important information. 4.1 Box-Ljung test The Box-Ljung test-statistics will be presented below: Table 2. Box-Ljung test Data: OMXS30 Return. X-squared = 2.4544 df = 1 p-value = 0.1172

From this Box-Ljung test, with a 90% significance level, we cannot reject the null hypothesis

that the data is iid and indicate serial-correlation. This is good since the Garch-models will fit

the data. Garch-model variables can be found in the appendix. Tome plot of the returns shows

that the data looks iid and heteroskedastic.

4.2 Forecast evaluation The statistical error measurements were used to see how far the estimations from the IV, Garch-

and Egarch-model were from realized volatility. Firstly, we start to see a pattern when we

measure the absolute return as a proxy for volatility. IV is significantly worse than the other

two models. The second-best forecast by a small margin is the Egarch model and the best

predictions come from the Garch-model when forecasting realized volatility proxies, see table

3.

Table 3. RV error measurement

Error measure IV GARCH EGARCH

MSE 0.007434753 0.001902215 0.001993558

MAE 0.07727785 0.03339234 0.03459173

RMSE 0.08622501 0.04361439 0.04464928

MAPE 7.003026 3.274548 3.424052

When comparing the forecasts with the proxy that is considered by earlier research as the best,

we find a different result. The IV still preforms the worst with the largest errors. Here we can

see that the EGARCH outperforms the GARCH models, see table 4.


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Table 4. IDRV error measurement

Error measure IV GARCH EGARCH

MSE 0.003163942 0.001529087 0.001361039

MAE 0.04836262 0.02566203 0.02345009

RMSE 0.05624893 0.03910354 0.03689227

MAPE 0.7964283 0.2865451 0.267878

To get a better visual understanding of the forecast compared to realized volatility I plot the

forecasts against the realized volatility.

RV = Black IV = Green GARCH = Orange EGARCH = Red

Looking at the RV estimate we can see that there are more fluctuations in the realized volatility

proxy than in any of the estimates. In this aspect, IV seems to be closer to the true changes in

volatility. The standard deviation of the variables can be found in table 5 below. IV seems too

often overestimate future volatility.


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IDRV = Black IV = Green GARCH = Orange EGARCH = Red

Looking at the better proxy for volatility, IDRV, we can see that all models seem to be more

accurate forecasting IDRV than RV. IV still seems to overestimate the volatility but looking to

follow the series better. This is to be expected since all the statistical error measurements were

smaller in forecasting IDRV than RV.

4.4 Summary of estimated variables

Table 5. Summary of estimated variables

Variable RV IV GARCH EGARCH IDRV

St. dev 0.04841062 0.03688079 0.02611346 0.02699393 0.04694509

Mean 0.05101191 0.124657 0.0680709 0.07118151 0.08336461

We can see clearly that the implied volatility measure overestimates the volatility for the period.

But it seems to follow the pattern of the series. The GARCH models seem to follow the series

better but still seem to overestimate the volatility.


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4.5 Correlation of estimated variables.

Table 6. Correlation between forecasts and realized values.

Correlation RV IV GARCH EGARCH IDRV

RV 1

IV 0.4781513 1

GARCH 0.5553162 0.7183221 1

EGARCH 0.5666410 0.7245656 0.9126824 1

IDRV 0.7706257 0.6177212 0.6432921 0.6776824 1

Here we can see that both of the realized volatility measures and the Garch-models forecast are

better correlated than the implied volatility. The Egarch-model is more correlated than the

simpler Garch-model. The IV is least correlated with both proxies for volatility measures.


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5 Discussion

In this section, I will analyze and discuss the results presented above, the strength and weaknesses of the modeling. I will continue to compare and analyze the results with previous studies and economic theory.

Looking at the results we can observe that Garch-models outperform implied volatility

forecasts. The implied volatility almost always predicts the volatility in the market to be greater

than realized volatility proxies. This result is not in line with Poon and Granger´s (2001) review

of 72 articles were most studies found that implied volatility was the better forecast for future

volatility. As they discussed this measurement is dependent on so many factors that will be

discussed below. On the OMXS 30 index during the period January 2016 to December 2018,

we can see that Garch-models outperformed implied volatility.

Information contained in IV, in theory, is the market participant´s expectations and forecasts

about future volatility. The GARCH-models information is from the historical data and only

tries to estimate the future. Therefore, in theory, IV should outperform Garch-modeling. In this

study, I have found that not to be true for the Swedish index OMXS 30.

Looking at studies by Antonucci (2008) and Kambouroudis, McMillan and Taskou (2016) who

found Garch-modeling to be superior to similar time-series modeling. The other study by

McMillan and Evans (2007) found that the Egarch-model performed the best in Sweden when

compared with similar time series models. This is in line with my result but in their study the

period of data was different.

In Sweden, the stock market for the assets included in the index is open 9:00 to 17:30 but the

derivatives market is open from 08:00 until 18:30. This indicates that options should have a

little more time to price more information in the market. Two and a half-hours in the aspect of

forecasting volatility in the short run can be determinative. Market specifics can be decisive

for the estimations and efficiency of volatility forecasting (Poon and Granger 2001).


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Poon and Granger (2001), Neely (2009) and Bakshi, Cho and Chen (2000) discussed how

market frictions can be an explanation for the difference predictability of implied volatility.

Sweden, in comparison, isn’t a large country an does not experience as large turnover, as in

larger markets like NYMEX. Consequently, resulting in non-continuous trading, relatively

large block-trades and alike. Market frictions were found to be a problem even in larger

markets. This can be a contributing factor to why IV was overestimating volatility. If an option

does not expedience a large turnover during a day the market won’t be able to approach the

equilibrium. The option can deviate from the Black-Sholes equilibrium because of these market

frictions, transaction cost and the bid-ask spread are often a reason for the asset not reaching

the Black-Scholes equilibrium. Therefore, the implied volatility can deviate from the true

market realized volatility. This can be an explanation for the difference in the forecasting ability

of IV when comparing to other studies on lager countries.

Many iterations of the Black-Scholes model have been researched. Green and Figlewski (1999)

investigate the error problems of the Black-Scholes model. It is possible that the standard

iteration of the Black-Scholes, I have used, is not used to price options in the Swedish market.

Firms and institutions that have found a better iteration of the Black-Scholes model will exploit

the market inefficiency and not share this information. This can be a contributing factor to the

result of IV forecasting the worst. I choose the model of Black-Sholes because of the principles

it is built on arbitrage opportunities and research indicating that the model still can be effective.

One part of this theses is the data, since its high frequent data I have a big data set. The data is

only during a booming economy and a short period. A similar result might not be attainable in

a crisis or recession. Unfortunately, because of the timetable of this thesis, I didn’t have time to

manage more data. The Garch-models are simple but stemming from previous research and

economic theory. Provided the result I believe the model choice used was ideal for this thesis.


26

Previously I mentioned that Black-Scholes builds on the arbitrage opportunities which are still

present. Market participants don’t always use the Black-Scholes model but rather similar or

variants of this formula. As discussed by Green and Figlewski (1999) the market participant

that is the writer of the option will be exposed to this model risk and that can be quite large

when writing options. The study by Neely (2009) investigated the volatility risk that investors

are exposed to when writing options. If an investor takes on risk, he will demand a risk

premium. This risk premium will be priced and added to the option price and reflected in the

expected volatility. Considering that this possibly is a contributing factor to the forecast of

implied volatility being larger than the realized volatility. Because the risks that stem from other

than the underlying assets price movements must be priced into the option valuation.

If we look at the correlation between the variable to be able to discuss the movements of the

measurements. We see that EGARCH-correlates the best with both volatility measurements.

The simpler GARCH-model correlates the second best with the volatility measurement. Lastly,

the Implied volatility correlates the worst with both realized volatility measures. The historical

data forecasting will produce better results than implied volatility from the options for OMXS

30 from 2016 to 2018.

One factor almost all research on Garch-modeling touch upon is the importance of accuracy in

the modeling distribution. My dataset follows the Student-t distribution quite well however with

fatter tails in the data than in the model from the Student-t. I do not consider this a problem,

although the importance of extreme values in returns often is overlooked when using time series

modeling. Looking at the study by Verhoeven and McAleer (2004) who found that kurtosis,

fat-tails, will not have a significant effect on return forecasting.

Looking at the results from we can see that on average the statistical error measures are

reasonably small. The EGARCH-models in particular can approximately forecast the volatility

of the OMXS 30. Even though the GARCH-model predicts marginally better on the RV

measure, there is a large difference in IDRV measure. The EGARCH-model will predict

volatility better since it accounts for leverage effects.


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6. Conclusion

In this section, I will conclude the results of my thesis and give an overview and feedback.

Finally, I will give concluding remarks and suggestions for future research.

The research questions were to analyze if I can forecast the volatility of the OMXS 30 index.

Evaluate what methods will provide the best forecast. I have analyzed and evaluated three

forecast models from economic theory and previous research. The models used were GARCH

and EGARCH on historical return data from the OMXS 30 index. The third model was Black-

Scholes implied volatility calculated from options with the OMX30 index as the underlying

asset to get implied volatility. The main goal was to get a better understanding of the forecast

ability, the measurements and economic theory in the field to better be able to forecast the

volatility.

The result shows that the EGARCH model is best for forecasting volatility on the OMXS 30

index. Implied volatility performed the worst and often overestimating future volatility. I find

multiple explanations for this overestimation. Examples of such are market frictions, model

error, not using optimized Black-Scholes-model and highlighting other risk factors that occur

when investing in options.

This study can be a framework for future research. If I had more time, I would have liked to

use a more complex Black-Scholes model on a longer dataset to see the performance of the

forecast in multiple states of the economy. There are more complex GARCH-models that use

the lavage effect differently than the EGARCH-model. It would have been interesting to

compare these models on a longer dataset.


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7 References

7.1 Books

Alexander, C., 2000. Market models. A guide to financial data analysis. John Wiley & Sonss

Ltd.

Cecchetti, S. G. and Schoenholtz, C. 2015. Money banking and financial markets. 4th ed.

McGraw-Hill Education.

Hull, J. C. 2012 Options futures and other derivatives. 8. Ed. Pearson education limited 2012.

Stock, J.H. and Watson, M.W. 2015. Introduction to econometrics. 3. rev. ed., Global ed.

Harlow: Pearson Education Limited.

Tsay, R. S. 2013. An introduction to analysis of financial data with r, Wiley.

Andersson 2000

7.2 Studies

Bashi, G., Cao, C. and Chen, Z. 2000. Do call prices and the underlying Stock always move in

the same direction? The review of financial studies fall, 13(3), pp. 549 – 584.

Bodie Z. and Merton R. C. 2009. The informational roll of asset prices, The case of implied

volatility. 1994. Working paper Harward Buisiness School.

Christensen B. J. and Parbhala, N. R. 1997. The relationship between implied volatility and

realized volatility. Journal of financial Economics, 50, pp. 125-150.

Evans, T. and McMillan, D. G. 2007. Volatility forecasts: the role of symmetric and long-

memory dynamics and regional evidence. Applied financial economics, 17, pp. 1421-1430.

Green, T. C. and Figlewski, S. 1999, Market risk and model risk for financial institutions

writing options. The journal of finance, 54(4), pp. 1465-1499.


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Hansen, P. R. and Lunde, A. 2005. A forecast comparison of volatility models: Does anything

beat the a Garch(1,1) Model? Journal of applied econometrics, 20, pp 873- 889.

Kambouroudis, D. S., McMillan, D. G. and Tsakou, K. 2016. Forecasting Stock return

volatility: A comparison of GARCH, Implied volatility and Realized volatility models. The

journal of Futures Markets, 36(12), pp. 1127-1163.

Ljung, G. M. and Box, G. E. P. 1978. On a Measure of lack of fit in the timeseries model,

Biometrica, 65, pp 297- 303.

Merton R. C. 1973. Theory of rational option pricing. Economics and management science,

4(1), pp. 141-183.

Neely, C. J. 2009. Dorcasting foreign exchange volatility: Why is implied volatility biased and

inefficient? And does it matter? Int. Fin. Markets, Inst, and Money, 19, pp. 188-205

Patton A. J. 2011. Volatility forecast comparison using imperfect volatility proxies. Journal of

Econometrics, 160(1), pp 246-256.

Poon S. H. and Granger C. 2001. Forecasting financial volatility, a review. Journal of Economic

Literature, 41(2), pp. 478–539.

Vee, D. Ng. C., Gunpot, P. N. and Sookia N. 2011. Forecasting Volatility of USD/MUR

exchange rate using Garch(1,1) model with GED and Student´s-t errors. University of Mauritus

research journal, 17, pp. 1-14.

Verhoeven, P. and McAller M. 2004. Fat tails and symmetric in finance volatility models.

Mathematics and computer in simulation, 64, pp. 351- 361.


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Appendix 8 Appendix 8.1 Autocorrelation function(ACF) and Partial Autocorrelation Function(PACF)


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8.2 Distribution of data

8.3 GARCH and EGARCH-models Table 8. GARCH-model estimates

Variable Estimate Std. Error t value Pr(>|t|)

omega 0.000000 0.000003 0.14346. 0,885930

alpha1 0.086554 0.121864 0.71024 0.477553

beta1 0.904124 0.105405 8.57759 0.000000

shape 6.516218 4.208410 1.54838 0.121531

Table 9. EGARCH-model estimates

Variable Estimate Std. Error t value Pr(>|t|)

omega -0.309490 0.004674 -66.2098 0.000000

alpha1 -0.162879 0.017250 -9.4423 0.000000

beta1 0.971346 0.000068 14329.2427 0.000000

gamma1 0.122843 0.016313 7.5303 0.000000

shape 7.800184 1.478620 5.2753 0.000000

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