IN DEGREE PROJECT MATHEMATICS,SECOND CYCLE, 30 CREDITS

, STOCKHOLM SWEDEN 2019

Volatility Evaluation Using Conditional Heteroscedasticity Models on Bitcoin, Ethereum and Ripple

DARKO BLAZEVIC

FREDRIK MARCUSSON

KTH ROYAL INSTITUTE OF TECHNOLOGYSCHOOL OF ENGINEERING SCIENCES

Volatility Evaluation Using Conditional Heteroscedasticity Models on Bitcoin, Ethereum and Ripple

DARKO BLAZEVIC

FREDRIK MARCUSSON

Degree Projects in Financial Mathematics (30 ECTS credits)

Master's Programme in Applied and Computational Mathematics (120 credits) KTH Royal Institute of Technology year 2019

Supervisor at KTH: Boualem Djehiche

Examiner at KTH: Boualem Djehiche

TRITA-SCI-GRU 2019:099

MAT-E 2019:55

Royal Institute of Technology School of Engineering Sciences KTH SCI SE-100 44 Stockholm, Sweden URL: www.kth.se/sci

Abstract

This study examines and compares the volatility in sample fit and out ofsample forecast of four different heteroscedasticity models, namely ARCH,GARCH, EGARCH and GJR-GARCH applied to Bitcoin, Ethereum andRipple. The models are fitted over the period from 2016-01-01 to 2017-12-31 and then used to obtain one day rolling forecasts during the period from2018-01-01 to 2018-12-31. The study investigates three different themesconsisting of the modelling framework structure, complexity of models andthe relation between a good in sample fit and good out of sample forecast.AIC and BIC are used to evaluate the in sample fit while MSE, MAE andR2LOG are used as loss functions when evaluating the out of sample fore-cast against the chosen Parkinson volatility proxy. The results show that aheavier tailed reference distribution than the normal distribution generallyimproves the in sample fit, while this generality is not found for the out ofsample forecast. Furthermore, it is shown that GARCH type models clearlyoutperform ARCH models in both in sample fit and out of sample forecast.For Ethereum, it is shown that the best fitted models also result in the bestout of sample forecast for all loss functions, while for Bitcoin non of the bestfitted models result in the best out of sample forecast. Finally, for Ripple,no generality between in sample fit and out of sample forecast is found.

Utvardering av volatilitet via betingadeheteroskedastiska modeller pa Bitcoin,

Ethereum och Ripple

Sammanfattning

Den har rapporten undersoker om battre anpassade volatilitetsmodeller ledertill battre prognoser av volatiliteten for olika heteroskedastiska modeller,i detta fall ARCH, GARCH, EGARCH och GJR-GARCH, med olika in-novationsdistributioner. Modellerna anpassas for Bitcoin, Ethereum ochRipple under 2016-01-01 till 2017-12-31 och darefter gors endagsprognoserunder perioden 2018-01-01 till 2018-12-31. Studien undersoker tre olikateman bestaende av modellstruktur, komplexitet av modeller och relatio-nen mellan en god passning och god prognos. For att evaluera passnin-gen for modellerna anvands AIC och BIC och for prognoserna anvandsforlustfunktionerna MSE, MAE och R2log som evaluering av prognosenmot den valda volatilitetsproxyn Parkinson. Resultaten visar pa att in-novationsdistributioner med tyngre svansar an normalfordelningen generelltleder till battre passning, medan man for prognoserna inte kan dra en sadanslutsats. Vidare visas det att GARCH-modellerna pavisade battre resultatbade for passning och prognoser an dem mer simpla ARCH-modellerna. ForEthereum var samma modell bast for samtliga forlustfunktioner medan Bit-coin visar olika modeller for respektive forlustfunktion. For Ripple kan inteheller nagon generalitet pavisas mellan passning och prognoser.

Acknowledgements

We would like to thank our supervisor Boualem Djehiche from the Depart-ment of Mathematics at KTH for insightful discussions, valuable input andmathematical guidance throughout the process of this thesis. Lastly wewould like to thank our families and friends for the support.

Contents

1 Introduction 11.1 Blockchain . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 41.2 Literature Review . . . . . . . . . . . . . . . . . . . . . . . . 5

2 Methodology and Data 82.1 Methodology . . . . . . . . . . . . . . . . . . . . . . . . . . . 82.2 Data . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9

3 Mathematical Theory 143.1 Stationarity and Financial Data . . . . . . . . . . . . . . . . . 143.2 Conditional mean . . . . . . . . . . . . . . . . . . . . . . . . . 153.3 Conditional variance . . . . . . . . . . . . . . . . . . . . . . . 15

3.3.1 Autoregressive Conditional Heteroscedasticity model(ARCH) . . . . . . . . . . . . . . . . . . . . . . . . . . 16

3.3.2 General Autoregressive Conditional Heteroscedastic-ity model (GARCH) . . . . . . . . . . . . . . . . . . . 17

3.3.3 Exponential General Autoregressive Conditional Het-eroscedasticity model (EGARCH) . . . . . . . . . . . 18

3.3.4 GJR General Autoregressive Conditional Heteroscedas-ticity model (GJR-GARCH) . . . . . . . . . . . . . . 18

3.4 Error distributions . . . . . . . . . . . . . . . . . . . . . . . . 193.5 Fitting and evaluation of in sample models . . . . . . . . . . 203.6 Evaluation of out of sample models . . . . . . . . . . . . . . . 21

4 Results 244.1 Optimal in sample fit . . . . . . . . . . . . . . . . . . . . . . . 24

4.1.1 Bitcoin . . . . . . . . . . . . . . . . . . . . . . . . . . 244.1.2 Ethereum . . . . . . . . . . . . . . . . . . . . . . . . . 264.1.3 Ripple . . . . . . . . . . . . . . . . . . . . . . . . . . . 27

4.2 Out of sample forecast . . . . . . . . . . . . . . . . . . . . . . 284.2.1 Bitcoin . . . . . . . . . . . . . . . . . . . . . . . . . . 294.2.2 Ethereum . . . . . . . . . . . . . . . . . . . . . . . . . 304.2.3 Ripple . . . . . . . . . . . . . . . . . . . . . . . . . . . 31

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4.2.4 Best performing forecasts for Bitcoin . . . . . . . . . . 324.2.5 Best performing forecasts for Ethereum . . . . . . . . 354.2.6 Best performing forecasts for Ripple . . . . . . . . . . 36

5 Analysis & Discussion 38

6 Conclusion 426.1 Suggestions for future research . . . . . . . . . . . . . . . . . 43

Appendices 46

A In sample results, Bitcoin 47A.0.1 ARCH(1) . . . . . . . . . . . . . . . . . . . . . . . . . 47A.0.2 ARCH(2) . . . . . . . . . . . . . . . . . . . . . . . . . 48A.0.3 ARCH(3) . . . . . . . . . . . . . . . . . . . . . . . . . 49A.0.4 GARCH(1,1) . . . . . . . . . . . . . . . . . . . . . . . 50A.0.5 EGARCH(1,1) . . . . . . . . . . . . . . . . . . . . . . 51A.0.6 GJR-GARCH(1,1) . . . . . . . . . . . . . . . . . . . . 52

A.1 In sample results, Ethereum . . . . . . . . . . . . . . . . . . . 53A.1.1 ARCH(1) . . . . . . . . . . . . . . . . . . . . . . . . . 53A.1.2 ARCH(2) . . . . . . . . . . . . . . . . . . . . . . . . . 54A.1.3 ARCH(3) . . . . . . . . . . . . . . . . . . . . . . . . . 55A.1.4 GARCH(1,1) . . . . . . . . . . . . . . . . . . . . . . . 56A.1.5 EGARCH(1,1) . . . . . . . . . . . . . . . . . . . . . . 57A.1.6 GJR-GARCH(1,1) . . . . . . . . . . . . . . . . . . . . 58

A.2 In sample results, Ripple . . . . . . . . . . . . . . . . . . . . . 59A.2.1 ARCH(1) . . . . . . . . . . . . . . . . . . . . . . . . . 59A.2.2 ARCH(2) . . . . . . . . . . . . . . . . . . . . . . . . . 60A.2.3 ARCH(3) . . . . . . . . . . . . . . . . . . . . . . . . . 61A.2.4 GARCH(1,1) . . . . . . . . . . . . . . . . . . . . . . . 62A.2.5 EGARCH(1,1) . . . . . . . . . . . . . . . . . . . . . . 63A.2.6 GJR-GARCH(1,1) . . . . . . . . . . . . . . . . . . . . 64

B Hypothesis tests 65B.0.1 Augmented Dickey-Fuller test . . . . . . . . . . . . . . 65B.0.2 Characteristics for Ethereum . . . . . . . . . . . . . . 66B.0.3 Characteristics for Ripple . . . . . . . . . . . . . . . . 69

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List of Figures

1.1 V olatility curves. . . . . . . . . . . . . . . . . . . . . . . . . 3

2.1 Log returns of Bitcoin. . . . . . . . . . . . . . . . . . . . . . 102.2 Squared log returns of Bitcoin. . . . . . . . . . . . . . . . . 102.3 QQ plot for Bitcoin plotted against the optimal normaldistribution 112.4 QQ plot for Bitcoin plotted against the optimal generalized

hyperbolic distribution . . . . . . . . . . . . . . . . . . . . . 12

3.1 Parkinson volatility estimator (red) vs Squared returns(black). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 23

4.1 Bitcoin forecast performances for used loss functions. . . 294.2 Ethereum forecast performances for used loss functions. 304.3 Ripple forecast performances for used loss functions. . . 314.4 Optimal forecast with MSE as loss function plotted against

the proxy. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 324.5 Optimal forecast with R2LOG as loss function plotted

against the proxy. . . . . . . . . . . . . . . . . . . . . . . . . 334.6 Optimal forecast with MAE as loss function plotted against

the proxy. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 344.7 Optimal forecast for all loss functions plotted against the

proxy. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 354.8 Optimal forecast with MSE as loss function plotted against

the proxy. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 364.9 Optimal forecast with MAE and R2LOG as loss functions

plotted against the proxy. . . . . . . . . . . . . . . . . . . . . 37

B.1 Log returns of Ethereum. . . . . . . . . . . . . . . . . . . . 66B.2 Squared log returns of Ethereum. . . . . . . . . . . . . . . 67B.3 QQ plot for Ethereum plotted against the optimal normaldistribution 68B.4 Log returns of Ripple. . . . . . . . . . . . . . . . . . . . . . 69B.5 Squared log returns of Ripple. . . . . . . . . . . . . . . . . . 70B.6 QQ plot for Ripple plotted against the optimal normaldistribution 71

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Chapter 1

Introduction

During 2017, the cryptocurrency market expanded as never seen before.With an exploding market cap, most cryptocurrencies reached new levels,with Bitcoin as the leading currency in media growing more than 1200%(Coinmarketcap, 2019). Following the peak in December 2017, most cryp-tocurrencies decreased by more than 50% compared to the peak. Eventhough the bubble burst, the lows of the crypto coin market cap have been in-creasing every year, suggesting a surplus capital inflow to the market (ibid).While it has been clear that the underlying technology blockchain has thepotential to reform the whole financial world, it is still hard to say what im-pact cryptocurrencies will bring to the table in the long run. An importantfactor for dealing with cryptocurrency is to gain an understanding of thedevelopment of the market in total as well as in particular currencies, since80% of the market cap consists of the ten biggest cryptocurrencies(ibid).The fact that more and more capital is invested in the cryptomarket foreach year implies the value of potentially predicting future movements.

As a part of being a currency, the initial purposes of cryptocurrencies wereto work as a medium of exchange, enabling lower costs and anonymizingtransactions via decentralized systems. During the last couple of years, theusage of cryptocurrencies seem to have changed path into a purely specu-lative investment for most users, leading to an explosive expand as a resultof traditional money being invested. As of today, the total market cap cir-culates around $128 billions, compared to $12 billions in the beginning of2016 (Coinmarketcap, 2019). As a result of the increasing interest, a needof quantifying the variation of cryptocurrencies becomes relevant. As statedabove, the price change of cryptocurrencies in general during the last coupleof years have resulted in a high volatility compared to traditional currencies.While different studies modelling volatility on traditional underlying, suchas exchange rates (Charef, 2017; Pilbeam Langeland, 2014 and Gao et al.,2012) and indices (Lin, 2018 and Sharma, 2015), little work exists on the

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cryptocurrencies volatility modelling both regarding fitting the in sampledata and forecasting the out of sample data.

A crucial and fundamental aspect in the world of finance is the study ofvolatility, which not too far ago was assumed to be constant in differentmodelling theories, lead by the most famous one performed by Black andScholes back in 1973. Nowadays, assuming constant volatility is replaced bythe knowledge of volatility being time-varying and predictable (AndersenBollerslev, 1997). No matter of asset being an exchange, index or stock,modelling and forecasting of volatility of returns is crucial in multiple set-tings within the finance sector, such as option pricing, where issuers arepricing the derivatives mainly dependent on the volatility of the underly-ing asset. Risk management and portfolio allocations are also exposed tothe future volatility in order to perform hedges and in estimating potentialrisks and losses, which have been more and more regulated according to law,lead by the MIFID (Markets in Financial Instruments Directive) regulations.

How are then volatility forecasts usually obtained? One common way toobtain the volatility is implied from the option market prices. Theoretically,this method contains all relevant information and parameters for estimatingthe future volatility via the Black Scholes model. However, the supportingevidence for this resulting in correct future volatilities could be questioned.Bollerslev and Zhou (2005) show that option prices in general contain a riskpremium due to that volatility cannot be perfectly hedged, which in generalgives an over-estimation of the volatility and thus higher option prices. An-other phenomenon suggesting questioning the volatility implied from optionpricing is the famous volatility smile obtained from the Black Scholes model.Figure 1 theoretically shows different strike prices of options with the same,arbitrary underlying and maturity date plotted against the volatility, show-ing that at the money options have minimum implied volatility, while inthe money calls and in the money puts in general have the highest impliedvolatility. Thus, the same market produces multiple implied volatilities forthe underlying in the same period. Additional to that, financial crisis haveresulted to a more skewed curve (Figure 1) in reality than the theoreti-cal smile curve obtained via Black Scholes due to the fact that institutionsstarted using out of the money puts in their hedging strategies, leading toan increase in value for these. Furthermore, the limited existing maturitydates in the options market limits the time horizon for volatility forecastwith this method, which causes problems for certain underlying assets, suchas cryptocurrencies, due to the very limited existence of options.

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Figure 1.1: V olatility curves.

Based on the above argumentation, this thesis will instead use the onlyobjective method of volatility forecast left which is available for all as-sets, namely time series modelling. Four different types of conditional het-eroscedasticity models, among many others, will be adapted to the cryp-tocurrency data and evaluated from two perspectives; in sample data (con-sisting of 730 data points between January 2016 to January 2018) and outof sample data (consisting of 365 data points between January 2018 andJanuary 2019). One of the first models enabling modelling conditional het-eroscedasticity in volatility was provided by Engle (1982), the Autoregres-sive Conditional Heteroscedasticity (ARCH) model, will be used. Althoughthe simplicity, the model requires many parameters in order to describe thevolatility process. As a consequence of this, Bollerslev (1986) developed theGeneralized Autoregressive Conditional Heteroscedasticity (GARCH) modelwhich is based on the precursor ARCH model, with the addition that far lessparameters are needed for modelling the volatility process. Studies (Ninget. al., 2015; Olbrys, 2012) have shown that innovations in financial as-set volatility have an asymmetric impact, which weakens both ARCH andGARCH models which take volatility clustering into account but assumethat both positive and negative shocks results in the same impact of thevolatility. This weakness was studied by Nelson (1991) who improved theexisting GARCH model to the Exponential Generalized Autoregressive Con-ditional Heteroscedasticity (EGARCH) model which enables to take into ac-count the differences between positive and negative asymmetric effects. Thefourth and final model used in this thesis is the Glosten-Jagannathan-RunkleGeneralized Autoregressive Conditional Heteroscedasticity (GJR-GARCH)model, which also models asymmetry.

This thesis aims to examine the four models adapted and evaluated onthe three largest cryptocurrencies from a market capitalization perspective,namely Bitcoin, Ethereum and Ripple. The data, existing of daily exchange

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rates against dollar for each cryptocurrency between January 2016 and Jan-uary 2019, was obtained from Coinmarketcap. Furthermore, the thesis aimsto examine the models in the cryptocurrency market in terms of in samplefit and out of the sample fit. The thesis will consist of three main themes.Primary, the structure of the modelling framework will be investigated, in-cluding different error distributions, leading to an understanding of how thisfactor affect the in sample fitting and out of sample forecasts. Secondly, anexamination between more and less sophisticated models will be obtainedin order to evaluate the potential differences regarding in sample and outof sample fit. The aim in this theme is not to evaluate the most accurateforecasts, but to evaluate if more complex models outperform models withless complexity. Finally, an examination in the relation between in samplefit and out of sample fit will be evaluated in order to determine if the betterin sample fit also results in a better out of sample fit.

This thesis contributes to existing literature by investigating volatility mod-els in a new cryptocurrency era, including both extreme upswing and down-swing, in contrast to previous research. The fact that the cryptocurrencyphenomenon is relatively new, the added time period of data in this thesiswill include a great amount of unexplored data. Also, to the best of au-thors’ knowledge, no volatility modelling has been performed and evaluatedfor Ripple so far.

1.1 Blockchain

The underlying technology enabling cryptocurrencies, blockchain, could bedescribed as a decentralized database which records transactions betweentwo parties efficiently and in a verifiable and permanent way. The processcould be described in a couple of steps starting with a transaction requestby the sending party. The request is then received by the network consistingof different computers(nodes) which are processing(mining) and validatingthe transactions before adding it to an existing blockchain, or declining it. Ifadded to a chain, the transaction is complete and permanent, leading to thedelivery being registered to the receiving party. In 2008, Satoshi Nakamoto,whose identity still remains unknown, introduced world’s first blockchainas a part of the first ever launched cryptocurrency, namely Bitcoin. Therevolutionary part to not require verification by a traditional third trustedparty raised the interest in cryptocurrencies. As of the end of 2016, thetotal market cap for cryptocurrencies exceeded $15 billions. The followingyear did not only attract regular payment users, but also investors andspeculators. As a result of this, the cryptomarket boomed to levels neverseen before, reaching a market cap all time peak of $830 billions. By theend of 2018 market cap bounced back and circulates around $125 billions.

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Until today’s date, over 2000 new currencies have been introduced to thecryptomarket. Although the amount of currencies being introduced, only ahandful represent the big majority of the market cap from year to year.

1.2 Literature Review

The following chapter will briefly present and explain previous literaturerelevant for the examination of the presented topic and research questions.The material used below is limited to only include peer reviewed articles.

In the study What good is a volatility model?, Engle and Patton (2001) dis-cuss GARCH modelling and forecasting where stylized facts about volatilityare highlighted. Primarily, it was shown that volatility exhibits persistence,meaning that large changes in price of an asset often is followed by otherlarge changes, and small changes in asset price leading to new small changes,which also was shown by Baillie et al (1996), Chou (1988) and Schwert(1989). The implication of all these studies regarding volatility clusteringshows that volatility shocks today significantly will influence many periodsof the future volatility. Secondly, Engle and Patton show that volatilityin the long run is mean reverting, meaning that there is a normal level ofvolatility to which volatility eventually will return. In other words, meanreversion in volatility implies that current information has no effect on thelong term forecast. Thirdly, the asymmetry of volatility is highlighted, alsoreferred as the leverage effect, which shows that negative shocks tend to havea greater impact than positive shocks on the volatility for equities. This re-flection is supported by other studies as well, such as Black (1976), Christie(1982), Nelson (1991) and Glosten et al (1993), where evidence of volatilitybeing negatively related to equity returns. Regardless of this, such evidencefor traditional exchange rates have not been found in general. Furthermore,the asymmetric structure of volatility generally generates a skewed distribu-tion of forecast prices, which was exemplified (Figure 1.1) for option pricing.

In the study Asymmetric V olatility in Cryptocurrencies, Baur and Dimpfl(2018) investigate 20 of world’s largest cryptocurrencies based on marketcapitalization. By using threshold GARCH (TGARCH) which takes asym-metry into account, they show a very different asymmetry effects comparedto the equity markets. According to the study, positive shocks increasevolatility by more than negative shocks. A suggested reason for this is ex-plained to be the so called ”fear to missing out” effect, where investors useto go long in the cryptocurrency market as a result of positive news. Thissuggested reason is showed to be stronger for Ripple than for Bitcoin andEthereum. Time series data for Ripple and Ethereum was from creationuntil 8 August 2018. For Bitcoin, times series data used was in the interval

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from 28 April 2018 to the same end date as for Ripple and Ethereum.

Bouoiyour et al. (2015) investigate nine different GARCH-models appliedon log returns of Bitcoin in two different time intervals. The optimal fitwas evaluated using AIC, BIC and HQC. In contrast to Baur and Dimpfl,the authors suggest that Bitcoin is likely to be driven by negative shocksrather than positive in both time intervals, where the degree of asymmetryis strong in the period between December 2010 to December 2014. As a re-sult of this, EGARCH, which accounts for asymmetry and leverage effects,gave the most optimal in sample fit in the used time interval. Furthermore,a shorter interval between 1 January 2015 to 20 July 2016 was examined,where the same asymmetry was spotted, even though is was weaker than inthe longer, previous period. Additionally, they also show that the durationof volatility persistence decreases drastically compared to the longer period.Finally, the authors conclude that the volatility seems to be on continueddecline since January 2015.

Bouri et al. (2017) present properties for Bitcoin before and after the pricecrash during 2013. Prior the crash, the results pointed towards a positiverelation between shocks and volatility, where positive news had a greaterimpact on volatility than negative. However, this property ceased in thepost crash period. The authors suggest that inverted asymmetry comparedto the equity market could be explained by the so called ”safe haven effect”,introduced by Baur (2011). In his study, Baur demonstrates the invertedasymmetry in gold, where gold is described to give rise to the safe havenproperty. According to this, investors interpret rising gold prices as a sig-nal for uncertainty in other markets, such as increased risk or uncertainty ofmacroeconomic and financial conditions. This on the other hand, introducesuncertainty in the gold market which yields to a higher volatility. Hence,positive shocks increase the volatility by more than negative ones.

GARCH Modelling of Cryptocurrencies (Chu et al., 2017) presents GARCH-type modelling of seven different cryptocurrencies, namely Bitcoin, Dash,Doecoin, Litecoind, Maidsafecoin, Monero and Ripple. Twelve differentGARCH models, namely SGARCH(1,1), EGARCH(1,1), GJRGARCH(1,1),APARCH(1,1), IGARCH(1,1), CSGARCH(1,1), GARCH(1,1), TGARCH(1,1),AVGARCH(1,1), NGARCH(1,1), NAGARCH(1,1) and ALL GARCH(1,1)were fitted to the log returns of cryptocurrency exchange rates where max-imum likelihood was used for fitting. Data between May 2017 and June2019 was used. The goodness of the in sample fit was evaluated in terms offive different criteria, consisting of AIC (Akaike information criterion), BIC(Bayesian information criterion), AIC (consistent Akaike information cri-terion), AICc (corrected Akaike information criterion) and HQC (Hannan-Quinn criterion), where smaller values indicated better fit. Eight differ-

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ent distributions (normal distribution, skew normal distribution, Student tdistribution, skew student t distribution, skew generalized error distribu-tion, normal inverse Gaussian distribution, generalized hyperbolic distribu-tion and Johnson’s SU distribution) of the innovations process were exam-ined, resulting in lowest value of all criteria for normal distribution for eachGARCH-type model and for each currency. Two exceptions were found,the first using TGARCH(1,1) applied to Ripple, and secondly using AV-GARCH(1,1) also applied to Ripple, where the innovation process followedskew normal distribution gave the best fit. Among the twelve best fittingmodels for each currency, the IGARCH(1,1) model with normal distributedinnovations resulted in lowest values of the criteria for five currencies, includ-ing Bitcoin. For Ripple, the GARCH(1,1) model with normal distributedinnovations resulted in lowest values of the criteria, and for Dogecoin, theGJRGARCH(1,1) model with normal innovations resulted in lowest values ofthe criteria. In the study V olatility estimation of bitcoin : A comparisonof GARCH models, Katsiampa (2017) also investigates the in sample fit ofsix different GARCH models (GARCH, EGARCH, TGARCH, APGARCH,CGARCH and ACGARCH) on Bitcoin log returns. Optimal models werealso chosen according to the three information criteria AIC, BIC and HQC.Results showed that the AR-GARCH model gave the most optimal in samplefit.

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Chapter 2

Methodology and Data

This paper will include four different volatility models introduced in theintroduction. The models will be applied on three different time series,namely Bitcoin, Ethereum and Ripple. In order to achieve the presentedaims, the so called in sample fit and out of sample forecast will be definedand motivated. Furthermore, some characteristics of the data sets will beillustrated.

2.1 Methodology

The set of data points will be divided into two subsets which will be referredas the in sample subset and the out of sample subset. We define the com-plete set consisting of S data points: p1, p2,..., pS . The in sample subset isthen defined consisting of p1, p2,..., pn, followed by the out of sample subsetcontaining pn+1, pn+2,..., pS . The in sample subset is evaluated via AIC andBIC, where two models (the best model according to each criterion) fromeach model family (i.e. ARCH, GARCH, EGARCH and GJR-GARCH)are selected. For the ARCH family, models with parameters combined ofq = 1, .., 3 are evaluated. For the GARCH models, the parameters are fixedto p = 1 and q = 1 but where the in-build ARMA model is tested for com-binations of p = 0, .., 3 and q = 0, .., 3, which is also the case for the ARCHmodels. This procedure is then obtained for four different error distributionsresulting in up to 72 models qualifying for the out of sample evaluation. Forthe out of sample forecast, the following iteration scheme is used:

Set the initial forecast origin to T = n. Fit each of the models to thein sample data containing p1, p2,..., pn. Select the best models in terms ofAIC and BIC for out of sample analysis.

Let h be the maximum forecast horizon, i.e. the last point of the initialout of sample subset. Calculate the one-step to h-step forecast using the

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best AIC/BIC fitted models.

Compute all forecast errors for each model, for each of the forecast stepsaccording to the definition below, where the actual volatility is determinedby a volatility proxy.

When this is performed, increase the origin by one step, which yields m =m+ 1 and repeat the process. This iteration should continue until the fore-cast origin m equals the last point in the original out of sample subset S

In order to evaluate the one-step to h-step forecasts some kind of loss func-tion must be introduced (see chapter 3). In the forecast scheme describedabove, the estimation sample increases as the forecast origin increases, whichimplies that all forecasts always are based on all available information. Eventhough the available information increases, in this study we will not re-fitthe model in order to limit the computational burden, which means thatthe models will be fitted only once to the original in sample subset. Thisstudy will only include the one day ahead forecast since different forecastinghorizons exceed the scope of this study.

2.2 Data

Data used in this study is obtained from Coinmarketcap, where the in sam-ple subset contains the daily opening price from January 2016 until January2018 and the out of sample subset from January 2018 to January 2019 forthe three biggest cryptocurrencies from a market capitalization perspective,Bitcoin, Ethereum and Ripple. According to Tsay (2008), a reasonablechoice is to set the in sample subset to 2

3 of the total set, and the out ofsample subset to the remaining 1

3 . When computing the volatility proxy,opening prices as well as daily high and low prices are used, see chapter 3.

In order to illustrate some characteristics for the data, different plots areused. Figure 2.1 shows daily log returns for Bitcoin, which visually seemsto be stationary with a mean around zero and also mean reverting, sincevolatility is moving between some range and not diverging. Clustering ofvolatility is also spotted, containing both relatively calm and turbulent pe-riods, which is one of the key characteristics of asset return volatility.

This is also shown when inspecting the daily squared log returns in Figure2.2, where relatively turbulent periods are followed by more calm periods

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Figure 2.1: Log returns of Bitcoin.

and vice versa.

Figure 2.2: Squared log returns of Bitcoin.

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Furthermore, the empirical distribution of the daily log returns is inves-tigated. Figure 2.3 shows a QQ plot for Bitcoin, containing the empiricaldistribution of the daily log returns plotted against the best fitted normaldistribution.

Figure 2.3: QQ plot for Bitcoin plotted against the optimalnormaldistribution

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Clearly, the empirical distribution of the daily log returns has heaviertails than the best fitted normal distribution, suggesting that another refer-ence distribution might be appropriate. Figure 2.4 illustrates the empiricaldistribution of daily log returns plotted against the best fitted generalizedhyperbolic distribution, showing a much better fit in the tails.

Figure 2.4: QQ plot for Bitcoin plotted against the optimal generalizedhyperbolic distribution

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The characteristics for Ethereum and Ripple are summarized in Ap-pendix A containing the corresponding plots as for Bitcoin. The character-istics for Ethereum and Ripple also follow the above description of Bitcoinwhere mean reversion, volatility clustering and heavy tails are identified.

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Chapter 3

Mathematical Theory

In this chapter we introduce some basic ideas of time series analysis andconditional variance models. As mentioned in the introduction, volatili-ties of asset returns are shown to be predictable and time-varying. Engleand Patton (2001) mention three important characteristics that should beconsidered. Firstly, volatility of returns show persistence, meaning thatvolatility shocks today will influence volatility expectations many periodsin the future. Secondly, regardless of its persistence, volatility is still meanreverting in the long run, meaning that after longer periods of higher orlower volatility, a correction towards the mean will occur. Thirdly, volatil-ity is asymmetric, meaning that negative shocks have greater impact thanpositive shocks. When modelling and forecasting volatility, theory suggeststhat the more of these characteristics are incorporated, the better descrip-tion of conditional variance will be obtained. Thus, the properties of eachpresented model below will be evaluated according to above.

3.1 Stationarity and Financial Data

A key role in time series analysis is played by processes whose properties, orsome of them, do not vary with time. If we wish to make predictions, thenclearly we must assume that something does not vary with time. In extrap-olating deterministic functions it is common practice to assume that eitherthe function itself or one of its derivatives are constant. The assumption ofa constant first derivative leads to linear extrapolation as means of predic-tion. In time series analysis our goal is to predict a series that typically isnot deterministic but contains a random component.

The closing price P on a trading day t of a particular underlying assetgenerally appears to be non-stationary. On the other hand, the log price ofthe underlying Yt := log(Pt) has observed sample paths, similarly like those

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of a random walk with stationary and uncorrelated increments, leading tothe log return of the underlying

Rt = Yt − Yt−1 = log(Pt)− log(Pt−1) = log(PtPt−1

). (3.1)

Furthermore, log returns have sample paths comparable to white noise. Al-though the comparison to white noise, strong evidence suggest that thesequence Rt is not independent (Brockwell Davis, 2016).

A model that is not stationary will not be mean reverting, which meansthat some shocks can potentially make the model diverge. Often the datathe models are built upon are also transformed to be stationary. There areseveral hypothesis tests developed to see if a time series model is stationary,the Dickey-Fuller test for example. In this thesis the log returns have beentested with the augmented Dickey-Fuller test (see appendix B) and has beenconfirmed stationary. The results are however omitted from the thesis.

3.2 Conditional mean

Conditional mean models are the most fundamental models used in time se-ries analysis. The ARMA(p,q) model uses past observations and innovationsto form a simple relationship with todays data.

E(Rt|Ft−1) = Rt = µ+φ1Rt−1+φ2Rt−2+...+φpRt−p+Zt+θ1Zt−1+...+θqZt−q(3.2)

where Zt is a WN(0, σ2) process.

The main benefits of the ARMA(p,q) is that good models can be createdonly from the dataset without the use of other explaining variables. In ad-dition, the models are more robust to changes in the datasets behaviour,compared to traditional regression models.

Financial time series often experience periods with low or extremely highvolatility, which is not captured by the ARMA(p,q) model. Therefore, thisreport will combine the ARMA(p,q) model with all of the heteroscedasticitymodels which will be explained, to see how different orders of p and q willbe able to capture the volatility of the data.

3.3 Conditional variance

The conditional variance V ar(Rt|Ft−1)=E((Rt − µt)2|Ft−1)=σ2t is modeledin a similar way as the conditional expectation. When combining ARMAwith conditional variance models, µ in the equation above is considered to

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follow an ARMA(p,q) model. There are several models that incorporate dif-ferent features such as giving larger volatility impacts to negative or positiveinnovations.

3.3.1 Autoregressive Conditional Heteroscedasticity model(ARCH)

Heteroskedasticity is a term describing that the variance of the data is chang-ing over time. This effect is very prominent in financial data as the econom-ical climate can change very rapidly. This can lead to periods where pricesrapidly grows or falls, resulting in very high volatility. The assumptionof traditional ARMA(p,q) models are that the variance is constant overtime, which can lead to under or over estimations when forecasting with themodels.To combat this the ARCH/GARCH models were developed, whichcombats the heteroskedasticity and leads to iid squared residuals.

The ARCH model introduced by Engle (1982) revolutionized the way tomodel conditional heteroscedasticity in volatility. The model itself is de-scribed as relatively simple compared to other models but requires manyparameters to describe the volatility. The ARCH(q) process Zt is defined asa stationary solution of the equations

Zt = htet, et IID(0, 1) (3.3)

where ht is the function of Zs, s < t, defined by

ht = α0 +

q∑i=1

αiZ2t−i (3.4)

with α0 > 0 and αj ≥ 0, j = 1, ..., q, implying a positive conditional vari-ance, where p is the order of the model. For ARCH models in general, theorder could be obtained from the Sample Partial Autocorrelation functionof squared returns (Sample PACF) where lags greater than q should be closeto zero. This usually results in high orders enabling accurate modelling ofthe conditional variance. Over time, it has been shown that higher ordermodels barely over perform lower orders in terms of out of sample volatil-ity forecast (Bollerslev et al., 1992). As a consequence of this, the order islimited to three in this study which also is favourable in terms of modellingcomplexity. Furthermore, the ARCH model manages to take volatility clus-tering into account, which can be identified in the definition of conditionalvariance, ht, where large Zt−i implies large ht. As a consequence of this,large shocks will be followed by large shocks and small shocks will be fol-lowed by small shocks. Even though volatility clustering is included in themodel, the previous mentioned asymmetry effects are not. This is easilyshown since the model only takes the shocks squared as an input. Another

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drawback is that the model is likely to over predict the volatility due to thefact that the model imposes restrictive intervals for parameters in order tohave finite fourth moments.

Consider an ARCH(p) model with j-steps forecast for h2k+j , which yields

h2k(j) = α0 +

p∑i=1

αih2k−i(j − i). (3.5)

This study will use the one-step forecast for some ARCH(q) model with amaximum order q of three, which yields the following one-step forecast

h2k(1) = α0 + α1h2k(0) + α2h

2k(−1) + α3h

2k(−2). (3.6)

3.3.2 General Autoregressive Conditional Heteroscedastic-ity model (GARCH)

The GARCH(p,q) process introduced by Bollerslev (1986) is a generaliza-tion of the above described ARCH(p) process. With similar properties, theGARCH process contains a modified variance equation defined according to

h2t = α0 +

p∑i=1

αiZ2t−i +

q∑j=1

βjh2t−j , (3.7)

with α0 > 0, αi ≥ 0, βj ≥ 0 and∑max(p,q)

i=1 (α0+βj)<1.Zt is, as in the ARCHmodel, defined by

Zt = htet, et IID(0, 1). (3.8)

Furthermore, the GARCH model requires less parameters for modelling thevolatility. This could easily be motivated by comparing equation 3.4 and3.7, where an additional term containing lagged conditional variances ht−jreduces the amount of lagged square returns Zt−i needed compared to theARCH model for volatility modelling. Similarly to the ARCH model, theGARCH model does not take asymmetry in volatility clustering into accountand also requires parameters to have a finite fourth moment. Consider aGARCH(p,q) model with j-steps forecast for h2k+j , which yields

h2k(j) = α0 + (α1 + β1)h2k(j − 1), (3.9)

where j > 1 and k is the forecast origin. A more explicit expression isobtained by substituting h2k(j − 1) iteratively, which yields

h2k(j) =α0(1− (α1 + β1)

j−1)

1− α1 − β1+ (α1 + β1)

j−1h2k(1). (3.10)

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3.3.3 Exponential General Autoregressive Conditional Het-eroscedasticity model (EGARCH)

The EGARCH model was introduced by Nelson (1991) developed from theprevious GARCH model. Unlike ARCH and GARCH, this model revolu-tionized the ability to distinguish the impact of positive and negative shocksin volatility clustering. As for ARCH and GARCH, Zt is assumed to beidentical according to

Zt = htet, et IID(0, 1), (3.11)

but with conditional variance properties according to

log(h2t ) = α0 +

p∑i=1

(αiZt−i + γi | Zt−i | −E(| Zt−i |)) +

q∑j=1

βjlog(h2t−j).

(3.12)In contrast to ARCH and GARCH, no restrictions for parameters are im-posed to avoid negative conditional variance. In order to highlight the asym-metry properties, a function f(Zt) is introduced where the magnitude effects(γ1) and the asymmetry effects (α1) from an EGARCH(1,1) model are pre-sented according to

f(Zt) = α1Zt−i + γ1(| Zt−i | −E(| Zt−i |)), (3.13)

where f(Zt) is uncorrelated and has zero mean due to the properties of Zt.Thus, equation (3.12) can be rewritten according to

f(Zt) = (α1 + γ1)Zt1(Zt > 0) + (α1 − γ1)Zt1(Zt < 0)− γ1E(| Zt |), (3.14)

The impact of positive and negative asset return effects are now easily seen,where positive shocks have an impact α1 + γ1 and negative shocks α1 − γ1.For the asymmetric effect, negative α1 implies greater impact from negativethan positive shocks. Thus, EGARCH is able to model volatility persistence,mean reversion and, unlike ARCH and GARCH, asymmetric effects.

3.3.4 GJR General Autoregressive Conditional Heteroscedas-ticity model (GJR-GARCH)

Glosten, Jagannathan and Runkle (1993) presented the GJR-GARCH modelwith similar properties as the previous explained EGARCH model. Ex-cept of being able to model volatility persistence and mean reversion, GJR-GARCH also models the asymmetrical effects. Zt is, as for the rest of themodels, defined according to equation (3.2). The conditional variance is nowdefined according to

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h2t = α0 +

p∑i=1

(αiZ2t−i(1− 1(Zt−i > 0)) + γiZ

2t−i1(Zt−i > 0)) +

q∑j=1

βjh2t−j

(3.15)with parameters α0 > 0, αi ≥ 0, βi ≥ 0 and γi ≥ 0 that guarantee a nonnegative conditional variance. In order to highlight the asymmetry prop-erties, a function f(Zt) is introduced where the magnitude effects (γ1) andthe asymmetry effects (α1) from an GJR-GARCH(1,1) model are presentedaccording to

f(Zt) = α1Zt−i + γ1(| Zt−i | −E(| Zt−i |)). (3.16)

Using the same reasoning as for the EGARCH model, f(Zt) can be writtenas

f(Zt) = (α1 + γ1)Zt1(Zt > 0) + (α1 − γ1)Zt1(Zt < 0)− γ1E(| Zt |). (3.17)

Thus, negative shocks have an impact α1 while positive shocks have animpact γ1. Consider a GJR-GARCH(p,q) model with j-steps forecast forh2k+j , which yields

h2k(j) = α0 + (α1 + γ1

2+ β1)h

2k(j − 1). (3.18)

where k is the forecast origin. A more explicit expression is obtained bysubstituting h2k(j − 1) iteratively, which yields

h2k(j) = α0

j−2∑i=0

α1 + γ12

+ β1)i + (

α1 + γ12

+ β1)j−1h2k(1). (3.19)

3.4 Error distributions

In all models used in this thesis, an error distribution must be set in orderto fully define each model. Equation 3.2 defines Zt, where the error termis presented as et. Regardless of which distribution it follows, et shouldhave certain properties, such as being identically distributed and indepen-dent with unit variance and zero mean. In this thesis, four different errordistributions are specified and used when modelling. As previous shown inthe descriptive data and in Appendix A, all three time series used displayrelative heavy tails compared to the best fitted normal distribution, whichmotivates that another choice of parametric family modelling heavier tailsmight be considered as a reference distribution instead. As a consequenceof this, three more distributions beyond the normal distribution are used,

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namely the student t distribution, the generalized error distribution and thegeneralized hyperbolic distribution. Following, all density functions for eachdistribution are defined.

Normal distribution density function:

f(z) =1

σ√

2πe−

(z−µ)2

2σ2 ,−∞ < z <∞ (3.20)

Student t distribution density function:

f(z) =Γ(v+1

2 )

Γ(v2√vπ

(1 +z2

v)−

v+12 ,−∞ < z <∞ (3.21)

Generalized error distribution density function:

f(z) =λs

2Γ(1s )e−λ

s|z−µ|s ,−∞ < z <∞ (3.22)

Generalized hyperbolic distribution density function:

f(z) =(γ/δ)λ√

2πKλ(δγ)eβ(z−µ)×

Kλ−1/2

(α√δ2 + (z − µ)2

)(√

δ2 + (z − µ)2/α)1/2−λ ×Kλ−1/2

(α√δ2 + (z − µ)2

)(√

δ2 + (z − µ)2/α)1/2−λ ,

(3.23)

−∞ < z <∞, where Kλ is the modified Bessel function of second kind.

3.5 Fitting and evaluation of in sample models

When fitting all mentioned models to the in sample data, all parameters areestimated for each model. The fitting will be obtained using the MaximumLikelihood Estimation method, and as a result of this, give the parametervalues for each model, for each distribution. When the models are obtained,each one of these will be evaluated according to the AIC and BIC.

Maximum Likelihood Estimation is a method of estimating the parametersof a statistical model so the observed data is most probable. The parametersare obtained by maximizing the likelihood function L(θ | z1, z2, .., zn) whereθ contains the set of parameters being estimated. The likelihood functioncould furthermore be described as the joint probability of the observed dataz1, z2, .., zn, which in this case is the in sample subset, over the parameterspace θ. The function is defined according to

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L(θ | Fn−1) =

n∏t=1

φ(Zt | Ft−1), (3.24)

where F denotes the available information at a specific time and φ the den-sity function for a certain distribution. In order to limit the computationalburden, the logarithm of the likelihood function will be used since maximiz-ing this is equivalent to maximizing the original likelihood function, whichyields to the following equation

logL(θ | Fn−1) =n∑t=1

logφ(Zt | Ft−1). (3.25)

As suggested in the literature review, two of the most used informationcriteria will be used, Akaike information criterion (AIC) and Bayesian in-formation criterion (BIC). These criteria will enable a way to compare themodel’s fit to the in sample data. For both of these, the maximum likelihoodwill be used together with a penalty based on the complexity of the modelaccording to below.

AIC = −2logL(θ) + 2k, (3.26)

where k is the number of parameters and logL(θ) the maximum logarithmiclikelihood function. For BIC, an additional parameter N is introduced whichequals the number of data points used in the in sample data, and is definedaccording to

BIC = −2logL(θ) + klog(N). (3.27)

The smaller value for a model, the better is the fit to the in sample data,including the penalty for the number of parameters and number of datapoints respectively.

To drive inference from our models they should be stationary as a non-stationary process can diverge and reach infinite variance. The model shouldalso have iid squared residuals, therefore an augmented Dickey-Fuller test isperformed on the residuals, as well as a weighted Ljung-Box test.

3.6 Evaluation of out of sample models

When all models have been determined in form of order and error distri-bution and fitted to the in sample data, followed by obtaining the one dayforecast for the whole out of sample period, the evaluation of these results

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is what remains. However, evaluating the performance of the forecastedvolatilities is far from trivial and standardized. The main reason might bethat conditional volatility itself is unobservable. As a consequence of this, aproxy for the actual volatility must be determined in order to obtain a refer-ence when comparing the forecasted volatilities. This section will highlightthe complexity of volatility and discuss different proxies considered, endingwith a specification for the proxy used.

It can be shown that squared returns is an unbiased estimator for the volatil-ity, however this proxy can be very noisy which resulted in that statisticiansfor a long time believed that GARCH models gave poor volatility predic-tions (Andersen and Bollerslev, 1998). Andersen and Bollerslev later foundthat depending on the proxy used GARCH models actually forecast volatil-ity well. Since then, several volatility proxies have been developed. TheParkinson estimator and the Garman-Klass estimator are two of the morefamous proxies. The former assumes constant trading and uses daily highsand daily lows to estimate the volatility. The Garman-Klass estimator alsoincludes opening and closing prices to provide a more accurate measure (San-tander, 2012). There are also other measures which account for opening gapsand drifts in the data. However, since the the cryptocurrency markets areopen nonstop, we believe that the Parkinson estimator will be well suitedfor cryptocurrencies. In addition, even though the Parkinson estimator isnot the best estimator in simulations, some studies have shown that it mightbe the best estimator on empirical data (Bennet, A. Gil, 2012). Figure 3.1shows the daily squared returns and the Parkinson estimator, applied to theEthereum. We can see that most of the time, the Parkinson estimator isless noisy, as mentioned in several articles.

σ2parkinson =ln(highlow )2

4ln(2)(3.28)

When the proxy is calculated, the models are evaluated using some of theloss functions discussed in (Bollerslev, Engle and Nelson 1993). In this study,the following loss functions will be used in order to evaluate the forecast ofthe different models.

MSE =1

n

n∑t=1

(σ2parkinson,t − σ2pred,t)2 (3.29)

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Figure 3.1: Parkinson volatility estimator (red) vs Squared returns(black).

R2log =1

n

n∑t=1

(log(σ2parkinson,tσ2pred,t

))2 (3.30)

MAE1

n

n∑t=1

|σ2parkinson,t − σ2pred,t| (3.31)

When choosing a loss function, the optimal one will always be the one thatpenalizes results which counter the goal of the model. For example, as men-tioned in Bollerslev, Engle and Nelson (1993), MSE might not be the bestmeasure, since it is fully symmetric and does not penalize zero or negativevariance. The R2log will exaggerate values close to zero and become largerif forecasts close to zero are wrong. MAE is similar to MSE in many cases,but it will not penalize large errors as much. West et al (1983) used a utilitybased measure which highlights the goal of the model, which in that case isto maximize risk adjusted returns, which should be the main factor whenchoosing a loss function. Since our goal is to study the behaviour of thedata, we chose these three loss functions above.

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Chapter 4

Results

This chapter will present the results obtained for each model. Primarily,the optimal resulting in sample fit will be presented, followed by the out ofsample forecasts. In order do limit the reading burden, all AIC and BICvalues are presented in Appendix A, meanwhile only the most optimal fit-ting models are presented in 4.1.

4.1 Optimal in sample fit

The optimal in sample fit will be presented for each cryptocurrency, start-ing of with Bitcoin, followed by Ethereum and Ripple. For each model, theparameters of the in-build ARMA(p,q) are specified together with the cor-responding AIC or BIC value.

4.1.1 Bitcoin

24

For the ARCH models, clearly, innovations with student t distribution yieldsthe most optimal fit. For the different GARCH models, the generalized errordistribution yields the lowest values of the criteria, with exception for theAIC for GARCH, where the generalized hyperbolic distribution is optimal.Furthermore, no matter of the model, the normal distribution results in thepoorest fit.

25

4.1.2 Ethereum

For all three ARCH models, innovations with generalized hyperbolic dis-tribution yields the most optimal fit. For all GARCH models, generalizedhyperbolic distribution yields the lowest AIC and generalized error distri-bution the lowest BIC.

26

4.1.3 Ripple

As for Ethereum, the most optimal fit for the ARCH models is obtainedusing student t distribution for Ripple. In contrast to both Bitcoin andEthereum, student t distribution also yields the lowest BIC for the EGARCHmodel and the lowest AIC for the GJR-GARCH model. For the differentGARCH models, generalized hyperbolic distribution also yields a good fit.

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4.2 Out of sample forecast

The best fitted (green marked from section 4.1) models are used to forecastthe volatility. In total, 24 models qualified. All of them are evaluatedusing three criteria, namely MSE, MAE and R2log as described in section3.10. The tables for each currency show the performance for each model,for each loss function. The most optimal model for each loss function andeach currency is highlighted. Furthermore, each of the highlighted modelsare illustrated by plotting the resulting forecast obtained by the model (redcurve) against the volatility proxy (black curve).

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4.2.1 Bitcoin

Figure 4.1: Bitcoin forecast performances for used loss functions.

29

4.2.2 Ethereum

Figure 4.2: Ethereum forecast performances for used loss functions.

30

4.2.3 Ripple

Figure 4.3: Ripple forecast performances for used loss functions.

31

4.2.4 Best performing forecasts for Bitcoin

For Bitcoin the GARCH(1,1)-ARMA(0,1) with generalized hyperbolic in-novations resulted in the most optimal forecast according to the MSE lossfunction. For R2LOG, the GJR-GARHC(1,1)-ARMA(0,1) with student t-distribution resulted in the best forecast and for MAE, GJR-GARCH(1,1)-ARMA(0,1) with generalized hyperbolic distribution resulted in the bestforecast. Below, forecasts from all models are plotted against the realizedvolatility proxy during the out of sample period.

Figure 4.4: Optimal forecast with MSE as loss function plotted againstthe proxy.

32

Figure 4.5: Optimal forecast with R2LOG as loss function plottedagainst the proxy.

33

Figure 4.6: Optimal forecast with MAE as loss function plotted againstthe proxy.

34

4.2.5 Best performing forecasts for Ethereum

For Ethereum the GARCH(1,1)-ARMA(1,2) with generalized hyperbolic in-novations resulted the best performing forecasts over all loss functions. Be-low, the obtained forecast is plotted against the realized volatility proxy.

Figure 4.7: Optimal forecast for all loss functions plotted against theproxy.

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4.2.6 Best performing forecasts for Ripple

For Ripple the GARCH(1,1)-ARMA(2,2) with generalized hyperbolic inno-vations resulted in the most optimal forecast according to the MSE lossfunction. Whereas the GARHC(1,1)-ARMA(1,0) with Generalized errordistribution resulted in the best forecast according to the R2log and MAEloss functions. Below, forecasts from both models are plotted against therealized volatility proxy during the out of sample period.

Figure 4.8: Optimal forecast with MSE as loss function plotted againstthe proxy.

36

Figure 4.9: Optimal forecast with MAE and R2LOG as loss functionsplotted against the proxy.

37

Chapter 5

Analysis & Discussion

In this chapter theory will be used to analyze and discuss the obtained re-sults for each cryptocurrency. The content will follow the presented themesand research questions, aiming for a foundation for the up-coming conclu-sion.

The first presented theme of this thesis is to investigate the structure of themodelling framework, where four different error distributions are included.When observing the in sample results, it is quite clear that the normal distri-bution results in a relatively poor fit, compared to the other ones. As shownin the introduction, the normal distribution could be questioned when heav-ier tails are present. When fitting the different ARCH models it is evidentthat the student t distribution outperforms all other distributions for Bit-coin and Ripple, meanwhile the generalized hyperbolic distribution is theoptimal one for Ethereum. For the different GARCH models, it is evidentthat the two generalized distributions give rise to the best fitted models forBitcoin and Ethereum. For Ripple, the optimal GARCH type models areobtained using the generalized error distribution or student t distribution.

For the out of sample forecast, different models are preferred when usingdifferent loss functions. This makes sense as they punish different charac-teristics of the model. When evaluating using a certain loss function, forinstance the MSE, we find that the generalized hyperbolic is the best dis-tribution for all the cryptocurrencies. When R2log is used, mixed resultsare obtained. For Bitcoin a t-distribution is preferred, for Ripple a gener-alized error distribution is preferred and finally for Ethereum a generalizedhyperbolic distribution is optimal. When evaluating using at the MAE, thegeneralized hyperbolic distribution is optimal for Bitcoin, generalized er-ror distribution for Ripple and generalized hyperbolic distribution is bestfor Ethereum. Both student t and the generalized distributions are classi-fied as heavy tailed distributions, which is described by Hult et al. (2012).

38

Over all, normal distribution, creates noisier models than the other distri-butions, both in and out of sample. The t-distribution fits reasonably welland many models are performing well in and out of sample. The generalizeddistributions can take the form of a distribution that closely resembles theinnovations, which in turn give us a model that better captures the data.Our results indicate that the normal distribution will not be suitable for thedata and will lead to worse forecast compared to more heavy-tailed distri-butions. As a contrast to the in sample results, the normal was the bestdistribution in terms of MSE in the ARCH models. In contrast to this, forthe GARCH models it was consistently the worst fit for R2log and on parwith the other distributions for the other loss functions.

The second theme of the thesis aims to examine if more advanced mod-els are better performing than less sophisticated models. As described inthe theory chapter, differences between the models are clear. Where one ofthe dimensions is the model itself, where the ARCH model is classified asthe least sophisticated one, meanwhile EGARCH and GJR-GARCH are themost sophisticated ones. The other dimension of complexity is the orderof the ARMA process (the sum of p and q), where a higher sum indicatesa more complex model. As can be seen in the out of sample results, astandard GARCH model is the best model for all loss functions for bothEthereum and Ripple, while GJR-GARCH is the best one for Bitcoin whenevaluating using R2log or MAE. Furthermore, when evaluating Bitcoin us-ing the MSE the GARCH model is again the optimal choice. Looking atall the ARCH models as a group we see that they perform worse no matterwhich loss function is used. As stated in the theory chapter, ARCH mod-els sometimes require a high order to describe the data accurately. Thisis seen in the out of sample results where the GARCH type models per-form a lot better as a group. When comparing model types in sample, wefind that the AIC/BIC for the ARCH models are worse compared to theGARCH/EGARCH/GJR-GARCH models for all cryptocurrencies. Whencomparing GARCH, EGARCH and GJR-GARCH models the AIC and BICare very close, this together with the out of sample results show us that theGARCH type models are a better choice compared to the simpler ARCHmodels.

The GARCH type however will not make as much of an impact in the inand out of sample, but it seems that most of the time the simpler standardGARCH model performs slightly better compared to the more advancedGJR-GARCH and EGARCH. Furthermore, the ability of taking eventualasymmetries in volatility into account seems not to give rise to better fore-casts. As shown in the literature review, the volatility asymmetry for cryp-tocurrencies have been changing during the past years in contrast to thestock market, where negative asymmetry is present. Since EGARCH and

39

GJR-GARCH do not perform better than the standard GARCH, one mightwounder if there actually exists asymmetry in volatility during the examinedperiod.

Regarding the number of variables, we can see that when choosing the op-timal model from the in sample data, AIC will prefer more complex modelsand the BIC will choose less complex models. For the out of sample period,we find that a less complex model often is preferred, where the highest ordermodel used has a sum of four while the overall median is one. This indicatesthat using a higher order ARMA process will not improve the model and itmay expose the researcher to over fitting the data.

The third and final theme of the thesis is to evaluate if a better in sam-ple fit also results in a better out of sample forecast. When comparing thein and out of sample results, it is evident that the overall best fitted model(standard GARCH) for Ethereum and Ripple also results in the best out ofsample forecast when using MSE as loss function. For Ethereum, this alsoholds for R2LOG and MAE as loss functions, while for Ripple it does not.For Bitcoin on the other hand, optimal fitted models do not result in thebest out of sample forecast for any of the loss functions which is contradic-tory to Ethereum.

When evaluating the most optimal in sample fit and out of sample fore-cast using MSE, it is observable that for specific model types separately(ARCH(1), ARCH(2), ARCH(3) etc.), the best in sample and out of sam-ple results do not necessarily correlate. For Bitcoin and Ripple, generallythe best in sample model does not perform best out of sample. Whereasfor Ethereum the opposite is the case, the best in sample model in generalleads to the best out of sample model for at least one of AIC or BIC. Wheninstead using R2LOG or MAE as loss function. the pattern for Bitcoin andRipple is the same as for using MSE, where the best in sample fit does notcorrespond do the best out of sample forecast in general. Using R2LOG andMAE as loss functions for Ethereum also yields a high correlation betweengood fit and good forecast.

What could then be the reason for an optimal in sample fit not resulting inan optimal out of sample forecast as the results indicate for Bitcoin and Rip-ple? One possible explanation to this could be the fact that the dynamics ofthe volatility might have changed during the in and out of sample periods.During the in sample period where the models are fitted, the cryptocurrencymarket showed extremely rising exchange rates in a typical bullish market.Even though the time period is relatively small, in total only three years,the huge drop of the whole cryptocurrency market precisely after the cutof point between in and out of sample periods gave rise to a bearish mar-

40

ket during the whole out of sample period. With an environmental changeof this size, it is quite likely that a change of market dynamics occurred.Thus, one could see this as a trade off problem, where an extremely goodfit might cause less flexibility to environmental changes such as volatilityshocks, leading to slow reactions.

Another thing to keep in mind is the volatility proxy. All results are com-puted against our proxy, the Parkinson volatility estimator. Thus anotherproxy, for example squared returns, would potentially result in a differentranking between the models. The choice of proxy is far from trivial, wherean alternative volatility proxy could have been squared returns which isunbiased. Although this, the squared returns proxy have also been shownto be very noisy and suboptimal in many cases. A potential proxy couldalso have been the more advanced Garman-Klass volatility estimator, butaccording to previous literature, no evidence could be found that this proxyoutperforms the Parkinson volatility estimator practically.

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Chapter 6

Conclusion

The following chapter will highlight the findings and conclusions of the ob-tained study. The structure will over again follow the three themes of thethesis in a chronological manner. Finally, suggestions for further researchwill be stated.

The investigation of the modelling framework structure shows that a normaldistribution clearly results in a relatively worse in sample fit compared tomore heavy tailed distributions no matter of heteroscedasticity model whenusing AIC and BIC as criteria. Clearly, it is evident that the empirical distri-bution from the used data exhibits significantly heavier tails than the normaldistribution, which motivates a reference distribution with heavier tails tobe more appropriate. For the out of sample forecast, a generally optimaldistribution is not evident. In contrast to the in sample conclusion, modelsusing normally distributed innovations are not clearly worse at forecastingthe volatility compared to the heavier tailed distributions. Furthermore,the choice of distribution seems to be dependent on the heteroscedasticitymodel used. Thus, there is no distribution that generally results in a betterforecast according to the used loss functions and volatility proxy.

When comparing more advanced models to less sophisticated ones, it is ev-ident that GARCH type models clearly outperform ARCH models in termsof in sample fit and out of sample forecast for all cryptocurrencies. It isalso evident that a lower order ARMA model for the conditional mean mostoften results in a better fit and forecast. Furthermore, no generality couldbe found

For Ethereum, it is shown that the best fitted in sample models resultsin the best out of sample forecast for all loss functions. For Ripple, the bestfitted in sample model only results in the best out of sample forecast whenusing MSE as loss function. For Bitcoin, non of the best fitted in sample

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models result in the best out of sample forecast.

6.1 Suggestions for future research

The most complex feature of volatility forecasting is the volatility proxy andloss functions used to evaluate the out of sample forecasts. A suggestion forfurther research could be to evaluate different proxies to investigate whetherit has an impact on how the models perform out of sample. For example howa more stable proxy such as the Garman-Klass volatility estimator changesthe ranking given the different loss functions. Maybe some loss functionsare more consistent to some volatility proxies.

During the time the models were fitted, cryptocurrencies were very noticedby the media, which potentially could have affected the volatility of thecryptocurrencies. Therefore, analysis during another period in which cryp-tocurrencies are not as noticed can potential yield other results.

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Bibliography

[1] Coinmarketcap Top 100 Cryptocurrencies by Market Capitalization.https://coinmarketcap.com, 2019.

[2] Charef, F. Non-linear causality between exchange rates, inflation, in-terest rate differential and terms of trade in Tunisia. Department ofFinance, Universite de Tunis El Manar, Tunis, Tunisia, 2017

[3] Pilbeam, K. and Langeland, K.N. Forecasting Exchange Rate Volatil-ity: GARCH models versus Implied Volatility Forecasts. Department ofFinance, City, University of London Institutional Repository, 2014

[4] Andersen, T.G., and Bollerslev, T. Heterogeneous information arrivalsand return volatility dynamics: Uncovering the long-run in high fre-quency returns.. Cambridge, Mass. National Bureau of Economic Re-search, 1997

[5] Bollerslev, T. Zhou, H. Volatility Puzzles: A Simple Framework forGauging Return - Volatility Regressions. Journal of Econometrics, 2005.

[6] Olbrys, J. Asymmetric Impact of Innovations on Volatility in the Caseof the US and CEEC–3 Markets: EGARCH Based Approach. -, 2012.

[7] Nelson, D.B. Conditional Heteroskedasticity in Asset Returns: A NewApproach. Econometrica, 1991.

[8] Bouri, E., Azzi, G, Dyhrberg, A.H. On the return-volatility relationshipin the Bitcoin market around the price crash of 2013. Economics, TheOpen-Access, Open-Assesment E-Journal 2017.

[9] Baur, D.G., Dimpfl, T Asymmetric volatilities in cryptocurrencies. Eco-nomics Letters, 2018.

[10] Baur, D.G. Asymmetric Volatility in the Gold Market. School of Financeand Economics University of Technology, Sydney, 2011.

[11] Katsiampa, P. Volatility estimation for Bitcoin: A comparison ofGARCH models. Economics Letters, 2017.

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[12] Chu, J., Chan, S., Nadarajah, S. Osterrieder, J. GARCH Modelling ofCryptocurrencies. Journal of Risk and Financial Management, 2017.

[13] Black, F. Studies in stock price volatility changes.. Proceedings of the1976 business meeting of the business and economics section, AmericanStatistical Association, 1976.

[14] Chou, R.Y. Volatility persistence and stock valuations: Some empiricalevidence using garch. Journal of Applied Econometrics, 1988.

[15] Schwert, G.W. Why Does Stock Market Volatility Change Over Time?.The Journal of Finance, 1989.

[16] Engle, R.F., Patton A.J. What good is a volatility model?. QuantitativeFinance, 2001.

[17] Baillie, R.T. Long memory processes and fractional integration ineconometrics. Journal of Econometrics, 1996.

[18] Kenneth D. West Hali J. Edison Dongchul Cho A Utility Based Com-parison of Some Models of Exchange Rate Volatility 1993

[19] Bollerslev, Engle, Nelson Arch models 1994

[20] Colin Bennet, Miguel A. Gil Measuring historical volatility 2012

[21] Lawrens R. Glosten, Ravi Jagannathan, David E. Runkle On the re-lationship between the expected value and the volatility of the nominalexcess return on stocks 1993

[22] Andrew A. Christe The stochastic behaviour of common stock variances:Value, Leverage and interest rate effects 1982

[23] Jamal Bouoiyour, Refk Selmi Bitcoin: A beginning of a new phase?2015

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Appendices

46

Appendix A

In sample results, Bitcoin

A.0.1 ARCH(1)

47

A.0.2 ARCH(2)

48

A.0.3 ARCH(3)

49

A.0.4 GARCH(1,1)

50

A.0.5 EGARCH(1,1)

51

A.0.6 GJR-GARCH(1,1)

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A.1 In sample results, Ethereum

A.1.1 ARCH(1)

53

A.1.2 ARCH(2)

54

A.1.3 ARCH(3)

55

A.1.4 GARCH(1,1)

56

A.1.5 EGARCH(1,1)

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A.1.6 GJR-GARCH(1,1)

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A.2 In sample results, Ripple

A.2.1 ARCH(1)

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A.2.2 ARCH(2)

60

A.2.3 ARCH(3)

61

A.2.4 GARCH(1,1)

62

A.2.5 EGARCH(1,1)

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A.2.6 GJR-GARCH(1,1)

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Appendix B

Hypothesis tests

B.0.1 Augmented Dickey-Fuller test

For the model

∆yt = α+ βt+ γyt−1 + δ1∆yt−1 + · · ·+ δp−1∆yt−p+1 + εt

assumes the null hypothesis H0: γ=0, against the alternative Ha: γ <0,where the test statistic is calculated according to

DFτ =γ

SE(γ).

The statistic is then compared to a Dickey-Fuller table

Ljung-Box test

The Ljung-Box test assumes under the null hypothesis that the data is iid,under the alternative hypothesis, it which assumes that the data exhibitscorrelation.

H0 : The data are independently distributed

Ha : The data are not independently distributed

The test statistic is calculated according to

Q = n (n+ 2)

h∑k=1

ρ2kn− k

,

where if Q > χ21−α,h the null hypothesis is rejected.

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B.0.2 Characteristics for Ethereum

Figure B.1: Log returns of Ethereum.

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Figure B.2: Squared log returns of Ethereum.

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Figure B.3: QQ plot for Ethereum plotted against the optimalnormaldistribution

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B.0.3 Characteristics for Ripple

Figure B.4: Log returns of Ripple.

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Figure B.5: Squared log returns of Ripple.

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Figure B.6: QQ plot for Ripple plotted against the optimalnormaldistribution

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