Comparable performance of GARCH-type models and implied volatility model on energy futures volatility estimation: Using crude oil futures as an example

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Master Thesis

Comparable performance of GARCH-type

models and implied volatility model on energy

futures volatility estimation: Using crude oil

futures as an example

Tianhan Ji S2910403 13 June 2016 MSc Finance University of Groningen Supervisor: Dr. Sibrand Drijver Keywords: crude oil futures, implied volatility,

GARCH, EGARCH, TGARCH, shock

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Acknowledgements

First and foremost, I would like to show my deepest gratitude to my supervisor, Dr. Sibrand Drijver, a very respectable and resourceful scholar, who has provided me with valuable guidance and patience in preparing and writing this report, especially in data collection and model processing. Without his enlightening instruction, I would not have known how to organize and complete my paper as I wanted.

I shall extend my thanks to all the professors, who gave me lectures about how to conduct researches and analyze data. Thanks to their willingness to provide me with their precious time I was able to get wonderful resources which enabled me to make a suitable data analysis.

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Declaration

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List of Tables

Table 1 Descriptive Statistics of WTI and Brent Crude Oil Futures Price Returns ... 14

Table 2 Autocorrelation Statistics ... 16

Table 3 Unit Root Tests ... 17

Table 4 Results from estimation of the model WTI Crude Oil Futures ... 23

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List of Figures

Figure 1 Logarithm of Oil Futures Price and Returns ... 15 Figure 2 Theoretical Quantile-Quantile ... 15 Figure 3 Dynamic forecasts of the conditional variance of WTI crude oil futures price returns based on GARCH model ... 18 Figure 4 Static forecasts of the conditional variance of WTI crude oil futures price returns based on

GARCH model ... 19 Figure 5 Dynamic forecasts of the conditional variance of WTI crude oil futures price returns based on EGARCH model ... 20 Figure 6 Static forecasts of the conditional variance of WTI crude oil futures price returns based on

EGARCH model ... 21 Figure 7 Dynamic forecasts of the conditional variance of WTI crude oil futures price returns based on TGARCH model ... 22 Figure 8 Static forecasts of the conditional variance of WTI crude oil futures price returns based on

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Summary

The paper focuses on the problem of volatility modelling in financial markets, using crude oil futures as an example. It begins with a general description of the properties of volatility, the usage and importance of volatility in financial markets (especially energy markets), and the contribution of this paper. The research is divided into two parts: estimation of conditional volatility based on Autoregressive conditional heteroskedastic family models (GARCH, EGARCH, TGARCH) and implied volatility model. The paper is focused on comparing the forecasting ability of different conditional volatility estimation model: implied volatility model (based on Black-Merton-Scholes Equation), GARCH, EGARCH and TGARCH.

Key words: crude oil futures, implied volatility, GARCH, EGARCH, TGARCH, shock persistence, volatility forecasting

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1. Introduction

Assets returns is one of the main characteristics of all financial assets, which is always considered to be a random variable and is impossible to know its future value with certainty (Hull, 2015). Since asset prices and asset returns change as a variable of interest, we are unable to observe the probability distribution of any asset prices or returns at a fixed moment of time. Normally we assume that the asset price follows Generalized Wiener Process: ∆𝑆 = 𝜇𝑆∆𝑡 + 𝜎𝑆𝜖√∆𝑡. The outcome of this variable is determined by the asset volatility. Volatility forecasting has been considered as one of the most important task in pricing financial assets and academics started to emphasize the importance of volatility forecasting since 1980s (Poon and Granger, 2003). All current literatures suggest that volatility forecasting modelling has a vital role in investment management, risk management, assets valuation and monetary policy making. With accurate forecasting of volatility of asset prices, it is easier to estimate the investment risk among the investment holding period, and thus the affect the Value at Risk and expected shortfalls.

It has been proven that financial commodity volatility’s behaviors are vital for derivative valuation, hedging construction, total marginal costs and opportunity costs of production of commodities (Narayan and Narayan, 2007). Fong and See (2002) explained in their paper that commodity prices tend to be extremely volatile over time. The volatile characteristic of commodity prices influences a lot in pricing commodity derivatives (futures, options, swaps) and constructing hedging portfolios. The findings confirmed the ‘choppy market’ definition suggested by Webb (1987): prices oscillate wildly without seeming to stop at intervening prices. Such volatile characteristics enhance the possibility of unexpected huge loss during trading due to the difficulties of predicting price trends and building hedging portfolios with optimal hedge ratio.

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and risk free interest rate, the implied volatility can be calculated based on the Black-Merton-Scholes Equation.

Given the importance of forecasting financial derivatives volatility, this research aims at providing volatility characteristics of crude oil futures, which is one of the most significant and classical energy futures in international futures markets. It is designed to compare different model types of estimating volatility of crude oil futures (WTI crude oil and Brent crude oil) from May 2011 to April 2016 and interpret the volatility clustering properties of crude oil futures through using theoretical and empirical analysis.

There are two main objectives of this research. First, it provides investment advice for investors. After the economic crisis in September 2008, participant numbers in energy futures market have showed a linear upward trend. In energy futures market, speculators accounted for the largest proportion; nevertheless, they often behave as individual subjects who show lack of information and blind investment, resulting in a mass of losses. This paper is based on the historical prices data of West Texas Intermediate (WTI) Crude Oil Futures and Brent Crude Oil Futures in Intercontinental Exchange (ICE), uses a variety of econometrics measurement methods to identify its volatility forecasting performances and functions and to study risk-benefit and volatility spillover effects on the two crude oil futures categories. By analyzing the WTI and Brent crude oil futures prices volatility, the report can find the optimal forecasting models for the futures prices volatility and provide advice for energy commodity futures investors. Second, this report aims to find the construction methods of energy futures market from the comparison of the two methods of volatility estimation. Through the study of optimal volatility prediction model of crude oil futures, we can understand the volatility clustering and prices fluctuations of futures prices among energy futures markets in the long-term predictions, analyze the degree of interaction and reasons for price fluctuations, find out the gap between different futures categories, and provide advice for investors of the construction of investing portfolios.

In order to meet these two goals, an implied volatility method from Black-Scholes Equation, GARCH, EGARCH and TGARCH model will be applied to estimate the crude oil prices volatility. This research uses the Black-Scholes Option Pricing Equation, GARCH model with an exogenous variable sequence, out-of-sample test to measure the forecasting ability and accuracy of different volatility forecasting methods.

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volatility in the long term over different types of data series and data frequencies, than other volatility forecasting models such as EWMA model and linear regression model. But generally no model can outperform all other models across different loss functions. In contrast, there is lack of literatures on understanding crude oil futures price volatility. This is very unfortunate due to the importance of crude oil futures among the energy markets, and the fact that the success or filature of a particular type of volatility forecasting model applied to one asset categories or one market cannot always apply to another assets or market. The related literature has managed to model oil price volatility (Narayan and Narayan, 2007); forecast volatility (Wei, Wang and Huang, 2010; Sévi, 2014); examined the relationship between crude oil price volatility and agricultural commodity markets (Du, Yu, and Hayes, 2011); discussed the factors that affect oil price volatility (Yang, Hwang and Huang, 2002).

This study concentrates on the performance among different futures prices volatility forecasting models in crude oil futures markets. The significance of the volatility forecasting contributes to have a sound global energy futures trading system, which can not only stabilize the international energy economic environment but also promote economic growth. Understanding the optimal volatility forecasting model of crude oil futures using ICE as examples can bring a fundamental comprehension about the futures market in the world. This thesis is mainly based on the literature from Agnolucci (2008), and Day and Lewis (1993). The added data from recent years makes the research results more valuable. We added TGARCH model in addition to GARCH(1,1) and EGARCH (1,1) model to test the effect of asymmetric information gained from downward market movements. Finally, a contribution is made towards the discussion on whether the specification of ARCH and GARCH terms adds value to the estimation. This study will address this issue by building GARCH model with generalized error distribution with ARCH and GARCH terms from 1 to 3. It is argued that GARCH(1,1) model suits better for most of the estimation and forecasting of financial assets volatilities. The improving of the GARCH forecasts indicates an increased futures and options energy market efficiency (shown as the higher liquidity) and yields different forecasting rankings (due to the reason of different time horizon).

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The remaining sections are organized as follows. Section 2 provides some preliminaries such as the volatility definition and measurement, literature review of the importance of forecasting crude oil futures volatilities, different volatility forecasting models, and build hypotheses based on the literatures. Section 3 introduces the three main categories of models that widely used in forecasting volatility, namely historical volatility model, implied volatility model (option implied standard deviation) and GARCH type models. On the basis of three main models, the forecasting performance evaluations are briefly introduced. Section 4 and 5 are the core sections of this research: section 4 discusses the data collection and research implementation, while section 5 explains the empirical results of the models and compares the performance of different forecasting models. Section 6 summarizes and concludes the findings of this paper, and provides some remaining directions for future research.

2. Theory and hypotheses development

2.1 Volatility definition and measurement

Volatility is one of the most important variables in pricing financial derivatives. For instance, when we price an option using Black-Merton-Scholes Equation, we need to know the volatility of the underlying asset from time now until the expiration time. Now we can even buy or sell financial derivatives which are written on volatility, as long as the definition and measurement of volatility is specified on the financial derivatives contracts clearly.

“A variable’s volatility, σ, is defined as the standard deviation of the return provided by the variable per unit of time when the return is expressed using continuous compounding.”(Hull, 2015). In definition, volatility is used to represent standard deviation, σ or variance, σ2, based the following formula: σ̂2₌ 1 N−1∑ (Rt− R̅) 2 N t=1 .

Where R_t represents the return at time t and R̅ represents the mean of returns. The sample standard deviation σ̂ is a distribution free parameter and only when there is a standard distribution such as a normal distribution, the cumulative probability density can be calculated analytically.

According to Figlewski (1997), the sample mean is biased a lot from the accurate estimate of the true mean, especially for small samples. To reduce the extreme situations, Figlewski suggested to take deviations around zero instead of the sample to enhance the volatility forecasting accuracy. In addition, the square root of σ̂2_{in the above equation is a biased estimate of true standard deviation due}

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Volatility is caused by the new information reaching the market the market whether the market is open for trading or not is irrelevant. When new information is available about the specific financial instruments or underlying assets, investors tend to revise their original opinions about the asset values. Hence, the value of assets changes, the price changes and it leads to the volatility. In short, volatility is due to the trading activities among the markets (Roll, 1984). Among all financial time series analysis, volatility analysis is one of the most vital aspects. For instance, option traders value volatility forecasting a lot for hedging decisions, in order to construct less risky hedging portfolios and gain more profits (Figlewski, 1997; Andersen & Bollerslev, 1998). Hence, the need of good analyzing and forecasting of volatility is increasing significantly.

2.2 The importance of crude oil futures and volatility

Energy products are among the most important and actively traded underlying commodities assets. Among those energy products, crude oil market is the largest commodity market in the world with an approximately 80 million barrels daily global demand (Hull, 2012). The international oil market is a capital intensive market deriving from numerous products types, transportation costs, storage difficulties, and especially, the tightening environmental protection issues. During the last two decades, crude oil has obtained the position of leading commodity futures in the world. According to Hull (2012), two important benchmarks for the crude oil futures prices are West Texas Intermediate (WTI) Crude Oil Futures and Brent Crude Oil Futures, so we use the two futures as sample for testing the volatility forecasting accuracy using implied volatility and Generalized Autoregressive Conditional Heteroscedasticity model.

Energy futures are of great importance of the real commodity markets and financial markets. Several empirical researches have showed that energy futures prices fluctuations have a significant influence on macroeconomic area, especially real economy variables, economic growth and economic inflation (Narayan & Sharma, 2013; Narayan et al., 2013).

Hamilton (1983) mentioned in his paper that during the post-OPEC phase, energy price increased rapidly and cruel oil plays a significant role in the world economy. The energy futures thus matter a lot to policy makers and investors. The global energy markets are constructed by high level of prices volatility and returns volatility clustering. Estimating volatility of futures can help in building macroeconomic models, pricing derivatives and reducing uncertainty in derivatives markets. For policy makers, the energy price and its volatility can be considered as the most important indicators when making policies among energy markets investments; while for investors, they matter on portfolio allocation and investment decisions (Lanouar, 2015). Engle (1982) examined the volatility of energy futures and came to a result that within energy series there exists volatility clustering property.

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derivatives and determining the optimal hedge ratio. Volatility estimation and forecasting thus are essential for effective financial derivatives pricing as well as effective and efficient managing and hedging market risks. The recent literatures show that researchers are interested in forecasting futures prices volatility, which matters a lot to Black-Scholes Option Pricing Model, CAPM model and Fama and French model.

2.3 Different methods of forecasting volatility

The literature of the methods of estimating volatility to date is large and diverse. There are several univariate and multivariate methods that can be used for estimating the futures volatility, for instance, forecasting model based on past volatilities (random walk model, historical average model, moving average model, exponentially smoothing model, exponentially weighted moving average model, linear regression model model, ARMA model); ARCH-type conditional volatility model (ARCH(q) model, GARCH (1,1) model, exponential GARCH model, threshold GARCH model, GJR-GARCH model, Quadratic GJR-GARCH model); Stochastic volatility models (Quasi-maximum likelihood estimation model); Options-based volatility forecasting model (Sadorsky, 2006). Four basic forecasting techniques for volatility are moving averages, exponential smoothing, linear regression and autoregression. In this paper, three main models will be used GARCH (1,1) model EGARCH model, and TGARCH model. Implied volatility model obtains the volatility from the inverting Black-Scholes Option Pricing Equation, GARCH-type forecasts time-series models to obtain volatility estimation data.

2.3.1 GARCH model

In typical literatures of volatility forecasting, researchers use GARCH-type models in energy commodity markets (Engle, 1982; Marzo & Paolo, 2010). Day and Lewis (1993) compared the forecasting accuracy of different methods of the volatility of crude oil futures prices. The research shows that in crude oil futures markets, both implied volatility in GARCH and EGARCH model’s explanatory powers are statistically significant but there’s no reason to believe that EGARCH performs better under the circumstances of asymmetric information of volatility responses. Sadorsky (2006) stated that the TGARCH model fits better for heating oil futures and natural gas futures volatility while the GARCH model are more suitable for crude oil futures volatility. And single equation GARCH model performs better than Vector Autoregression model and bivariate GARCH in predicting the futures volatility.

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the volatilities estimation of individual assets, the multivariate models perform better, while univariate models show better results for crack spread (the difference between crude oil price and petroleum products price which extracted from it) volatility due to asymmetric effects. According to Murat and Tokat (2008), crack spread is referring to the crude oil- product price relationship and crack spread futures tend to be extremely violate over time. Thus, univariate GARCH models are adopted to estimate the volatility for crack spreads.

Klaassen (2002) improved single regime GARCH model with Markov regime-switching model to generate volatility forecasts, due to the volatility persistence. The regime-switching model shows a significantly better out of sample forecasting. Fong and See (2002) defined volatility regimes by major events affecting supply and demand for crude oil. Their out-of-sample tests showed that the Markov regime switching GARCH model performs significantly better than single-regime GARCH models due to the evolution of volatility. And it provided an interesting framework for investors for short-term energy futures volatility forecasting. Based on the previous literatures, Lanouar (2015) used the Markov Switching GARCH model by probability smoothing to compare the estimated long range dependence in energy futures prices volatility. In terms of Value at risk performances and out-of-sample forecasting, the FIGARCH and FIEGARCH models perform better than Markov switching GARCH model. Cifarelli and Paladino (2015) also contributed to the literatures of Markov regime-switching model by design a two-stage regime-regime-switching model to capture the hedging and speculating features of futures markets and define risk appetites.

Agnolucci (2009) compared the implied volatility model with GARCH model to show the predictive ability of those two models to cruel oil futures volatility. Based on the WTI future contracts, the GARCH-type models show a better predictive accuracy than implied volatility from the Black-Scholes Option Pricing Equation due to the asymmetric information existing in the markets. However, there exists some information in implied volatility forecasting while not in GARCH-type models. And the GARCH (1,1) is the best forecasting model in WTI crude oil futures compared to other ARCH and GARCH terms.

2.3.2 Implied volatility model

In addition to the volatility approach (conditional variance), implied volatility estimates can forecast volatility on the basis of the observed market prices of options of underlying assets. Implied volatility sometimes biased from the realized volatility, thus, it is a forward-looking estimation method, compared to the backward-looking models like GARCH models which based on the historical returns data.

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contains only past information in current data that already included in implied volatility, but no economically predictive information, which indicates that implied volatility is generally outperformed in a relative sense. Wu, Myers, Guan and Wang (2015) continued the implied volatility method and used it on predicting realized volatility in corn futures prices. They improved the implied volatility with risk-adjusted and figured out that risk-adjusted volatility is more efficient and unbiased than the original volatility model due to the model-free implied volatility is related to risk premiums.

3. Methodology

3.1 Hypotheses Hypothesis 1:

GARCH-type models show better performance than the implied volatility models in asymmetric information among the crude oil markets.

Hypothesis 2:

The increasing ARCH and GARCH terms does not provide additional value to the estimation and forecasting accuracy in GARCH models.

Hypothesis 3:

GED distribution in GARCH model provides a better performance of the fat tail returns on crude oil futures, compared to normal error distribution and student’s t error distribution, as suggested by Xu and Taylor for currency futures in 1994.

Note: GED distribution refers to version 1 of generalized normal distribution or generalized Gaussian distribution. It includes all the characteristics of normal and Laplace distribution, as well as all continuous uniform distributions on real line bounded intervals. In this case, we use normal distribution with estimation of 𝛽 = 2 (mean µ and variance 𝛼2

2). Since the crude oil futures statistics

show a fat tail characteristics, we assume 𝛽 < 2 to allow heavier tails. Hypothesis 4:

EGARCH and TGARCH models perform better than GARCH (1,1) model under the circumstances of asymmetric information of volatility responses.

3.2 Methodology of data collection

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market. Besides, every contract has maturity date, which cannot represent the long-term variation. In particular, the monthly data and the yearly data are not representative enough to analyze the relationship . Thus, daily closing price data will be most suitable indicator to describe the target population.

In order to overcome the in-continuity problem of futures contracts, and make the sample typical, we use the consecutive daily price of crude oil continue futures to do the research for reflecting the tendency of the crude oil future prices. To keep comparability, we use the same methods to collect and process the data for WTI and Brent crude oil futures. The census takes advantage of counting all elements in a population, which will be more desirable than the sampling method in the research. Thus, the target population is the daily data in those presumed variables above and the population size is the size of sample since census method is inserted in the study.

According to the characters of futures markets, the futures contract price is in-continuity. As the futures contracts coming to the maturity, the futures price will be fluctuated in a large extent, which causes problems for data reduction and analysis. Thus we will delete the data of the nearest ten days to maturity, in order to delete the effect of trading days; however, the method still has some problems. All of the data processing is operated by Eviews, if there is not some special mention.

3.3 Preliminary Analysis

The empirical research can be divided into three main phases: the first phase deals with some descriptive features of the futures prices and returns and conducts several tests to identify the existence of volatility in the data. The second phase is dealing with different methods of forecasting volatility: historical volatility, implied volatility and GARCH type models. The model is selected by the model selection criteria such as Akaike Information Criterion (AIC), Schwartz Information Criterion (SIC), and Hannan-Quinn Information Criterion (HQC). The third phase uses several tests to identify the best fit model of forecasting volatility, especially out-of-sample test.

The preliminary analysis should be done in three areas: firstly, the descriptive statistic for crude oil futures prices and its returns, including the figures, secondly the ARMA(p,q) model for the two financial time series, and lastly an ARCH test will be used to test the existence of volatility in the oil futures price returns.

We begin with the transformation of futures price to return on the futures price of crude oil futures 𝑅_𝑡: 𝑅_𝑡 =(𝑃𝑡−𝑃𝑡−1)

𝑃𝑡−1

,

where 𝑃𝑡 is the crude oil futures daily settlement price. We choose returns

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return as a logarithm of price at the end of the day minus the logarithm of price at the end of previous day, which is 𝑟𝑡 = ln(1 + 𝑅𝑡) = ln⁡(_𝑃𝑃𝑡

𝑡−1).

We then take the logarithm of futures prices as random walk process, as the crude oil futures returns are assumed to be normally distributed: where the 𝜀_𝑡 represents the errors.

𝑃_𝑡 = 𝛼 + 𝑃_𝑡−1+ 𝜀_𝑡 𝑟_𝑡 = 𝑃_𝑡− 𝑃_𝑡−1= 𝜇 + 𝜀_𝑡

3.4 Model building 3.4.1 GARCH model

Tim Bollerslev (1986) suggested the standard linear GARCH (p, q) model for the crude oil price returns, which shows below:

ln(σ_t2_{) = ϖ + ∑ α}

𝑖ut−i2 + ∑ β𝑖ln(σt−i2 )

The current variance depends on the first p past conditional variances and the q past squared innovations. Considering the information of lags, there’s usually only one lag term t-12

in the GARCH models. The GARCH (1,1) is shown below:

ln(σ_t2_{) = ϖ + αu} t−i 2 _{[1 − 𝐼} {ut−i>0}] + ξut−1 2 _𝐼 {ut−i>0}+ βln(σt−1 2 ₎

where I_t−i = {1, 𝑖𝑓u_{0, 𝑖𝑓u}t−1> 0;

t−1 ≤ 0.

I_t−i⁡ becomes an indicator variable when the previous lagged residual term is positive. So the GARCH (1,1) can be rewrite as: ln(σt2) = ϖ + αut−i2 + βln(σt−12 ) (Klaassen, 2002). ut−i2 is the

lagged squared residual term (ARCH term) and ln(σt−12 ) is the lagged conditional variance term

(GARCH term). GARCH (1,1) is the simplest form, where the conditional variance at time t not only depends on the lagged squared residual term at time t-1 but also on the lagged conditional variance term at time t-1. GARCH (1,1) model has less parameters than ARCH models which indicates a lower chance of a negative parameter.

When estimating the GARCH (1,1) model, we estimated all three cases of normal error distribution with heteroskedasticity consistent covariance- Bollerslev & Wooldridge, Generalized error distribution and student’s t error distribution. The conditional normality assumption doesn’t capture the property of fat tail. As shown in the preliminary analysis. Unlike normal error distribution, GED and T distribution allow for fat tails in the conditional distribution.

3.4.2 EGARCH model

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than the same magnitude of positive shocks, due to the risk adverse attitudes of investors. Nelson (1991) introduced the Exponential Generalized Autoregressive Conditional Heteroscedasticity model (EGARCH) which builds a directional relationship of financial assets price moves on the movements of conditional variance (volatility). The large price downward movements have a larger impact on volatility than the large price upward movements. The EGARCH (1,1) model formula is shown below:

ln(σ_t2_{) = ϖ + βln(σ} t−1 2 _{) + γ} ut−1 √σ_t−12 + α [ |ut−1| √σ_t−12 − √ 2 π] [|ut−1| √σ_t−12 − √ 2

π] is the lagged squared residual term (ARCH term), ut−1

√σ_t−12 is the asymmetry term

and ln(σ_t−12 _{) is the lagged conditional variance term (GARCH term). If γ parameter is negative, then}

the model indicates a higher effect from negative shocks than positive shocks on financial assets volatilities. In accordance with the empirical evidence from stock markets returns, the conditional variance tends to increase or decrease when u_t−1 is negative or positive. However, the logarithm form of the volatility causes difficulties when we have to estimate an unbiased forecast (Bollerslev, 2009).

3.4.3 TGARCH model

Glosten Jagannathan, and Runkle (1993) suggested the Threshold GARCH (TGARCH) model (which also known as GJR-GARCH model) to eliminate the asymmetric influences of upward and downward movements towards the market. The variance equation is:

ln(σt2) = ϖ + αut−i2 + δDt−iut−12 + βln(σt−12 )

or ln(σ_t2_{) = ϖ + (α + δD}

t−i)ut−12 + βln(σt−12 )

where D_t−i= {1, 𝑖𝑓u_{0, 𝑖𝑓u}t−1 < 0;

t−1≥ 0.

D_t−i⁡will be an indicator variable if the previous lagged residual term is negative. δ parameter can eliminate the influence of the using of maximum likelihood techniques. It is similar to the GARCH model mentioned before: ln(σt2) = ϖ + αut−i2 [1 − 𝐼{ut−i>0}] + ξut−1

2 _𝐼

{ut−i>0}+ βln(σt−1

2 _).

The only difference between the GARCH and TGARCH models is that in the TGARCH model, the lagged squared residual term will have larger effect on the conditional volatility if the lagged residual term u_t−1 is negative, due to the reason that δ will be have a positive value (Tsay, 2005).

3.4.4 Implied volatility

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The implied volatility (IV) is differing from the historical volatility that it should reflect but the market’s expectations. The crude oil futures all have correspondent options, which all inputs to the Black-Scholes Equation are observable in the markets. According to B-S model, the volatility (also referred to constant instantaneous variance) can be calculated using the market prices of the options. It defines an implied volatility forecasting of the underlying asset prices (which referred to crude oil futures in this case) during the remaining life of the options.

We use the arithmetic average from both the call and put options of crude oil futures to increase the accuracy of the estimator and reduce the observation errors. The time series is obtained from the times series of nearest-to-the money options of the nearest-series with strike price just above the underlying futures price and the time series of nearest-to-the money options of the nearest-series with strike price just below the underlying futures price. According to Agnolucci (2009), the nearest-to-the-money series has the shortest maturity time (which no shorter than eight calendar days).

To produce a Black-Scholes model for the options price, the input parameters are: current stock price S0, option strike price K, risk free interest rate rf, time to maturity of the option T, standard

deviation of the returns on the underlying assets 𝜎. We reverse the Black Scholes model to obtain 𝜎 from the other five variables. 𝜎 here refers to option implied standard deviation (ISD). Based on the assumption of Black-Scholes model, the volatility is constant. However, Merton (1973) proved that ISDs differ from each other due to the different maturity of the options, so the 𝜎 generated from the formula might be biased. Bodie and Merton (1995) then suggested that for nearest-to-maturity options, the biased of the ISD will be minimized by selecting the nearest-to-the-money options.

The data we obtained from Datastream gave directly the IV (refers to implied volatility), so there’s no need to employ Black-Scholes model. However, the pricing formula used the calendar days while IV is based on trading days: 𝜎𝑡 = 𝜎𝑐(√𝑇_√𝑇𝑐

𝑡) where 𝑇𝑐 is the number of calendar days 365 and 𝑇𝑡

is the number of trading days 250. According to Fleming (1995), 𝑇𝑡 = 𝑇𝑐− 2 ∗ 𝑖𝑛𝑡 (𝑇₇𝑐⁡) = 261, which

shows better performance the the trading days number we used in pricing formula. We choose the option whose time to maturity is 28 calendar days (for 4-week forecasts).In order to conducting the implied volatility regressions, we divided the IV series obtained by the √261, the average number of trading days in one year, and then get the volatility series from squared IV.

3.5 Post-estimation forecasting and model comparison

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Mean logarithm of absolute errors (MLAE) and Theil-U statistic. Those measures can be obtained directly from Eviews forecasting analysis.

4. Data and Descriptive Analysis

4.1 Data

This research concentrates on the volatility forecasting ability for crude oil futures of implied volatility and GARCH type models. There is no available instrument to measure those macroeconomic variables above. Consequently, all data are collected from the secondary data resource which is too comprehensive to stand for population. Ideally, to eliminate the extreme cases of financial crisis and lead to a valid volatility forecasting model, it is the best to collect the data from 2011 after the financial crisis. In particular, yearly and monthly closing prices data are less representative. Thus daily closing prices data will be the most suitable indicator to describe the target population. The census takes advantage of counting all elements in a population, which will be more desirable than the sampling method in the research. Thus, the target population is the daily closing data in those presumed variables above and the population size is the size of sample since census method is inserted in the study.

All data are collected from DataStream database, where daily closing prices of Brent and WTI crude oil futures are recorded in Dollars. By analyzing the trend of futures prices of WTI Crude Oil Futures and Brent Crude Oil Futures, it is obviously that the original price is non-stationary. So we will transform both data of price and volume into logarithmic form in order to reduce the variances. This research uses daily closing price of Intercontinental Exchange (ICE) West Texas Intermediate (WTI) Crude Oil Futures and Brent Crude Oil Futures as raw data. The daily data of WTI futures cover from 5 September 2011 to 29 April 2016 for a total of 1215 observations and the daily data of Brent futures cover from 27 May 2011 to 21 April 2016 for a total of 1280 observations. The data used in this study are daily returns on the nearest crude oil futures based on the WTI and Brent crude oil futures traded on ICE. We use daily prices for the nearest futures contract to proxy for the spot price of the crude oil. Daily returns are computed by taking the difference in logarithm of consecutive days’ closing prices, i.e. 𝑟_𝑡 = 𝑙𝑛 𝑝𝑡

𝑝𝑡−1

4.2 Contract specifications

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and maximum price changes are $0.01 and $15 per barrel. The maturity of WTI and Brent crude oil futures is on the third business day before the 25th calendar day of every month.

4.3 Preliminary analysis

We plot the logarithm of daily crude oil futures settlement prices and daily crude oil futures prices returns in Figure 1. Figure 1 shows that the futures returns have the characteristic of volatility clustering (the periods of high volatility are followed by the periods of tranquility), and also structural breaks in volatility. There is lack of evidence to support a normally distribution series (which can also conclude from the statistics, Jarque-Beta test in Table 1). Due to the volatility clustering property, the returns of crude oil futures are not identically distributed with zero mean and constant variance σ2. The conditional variance changes over time, which refers to heteroskedasticity. In Figure 2 we plot the Theoretical Quantile-Quantile graphs for the logarithm of oil futures price and oil futures price returns, respectively. The figure shows that both large positive and negative shocks contribute to the non-normality properties of the WTI and Brent time sequences.

Table 1 shows the descriptive statistics of the two time series data. The means of daily futures return are extremely small (0.05% and 0.07%) compared to the standard deviation (2.13% and 1.94%). The test statistics, such as the Jarque-Bera test, suggests that for the two crude oil futures, neither the prices series nor the returns series are normally distributed. Both of the crude oil futures price returns display some evidence of kurtosis and skewness. The skewness and kurtosis figures show that the unconditional distribution of the returns is skewed (compared to normal distribution skewness of 0 and kurtosis of 3) and consistent with many other financial assets. Both of the series are skewed towards the right side and have distributions with tails that are fatter than a normal distribution. The statistics from Jarque-Bera Test shows that both series are distributed non-normally.

Table 1 Descriptive Statistics of WTI and Brent Crude Oil Futures Price Returns

WTI Brent Mean -0.000521 -0.000720 Median -0.000285 -0.000556 Maximum 0.119132 0.104162 Minimum -0.090703 -0.088574 Std. Dev. 0.021257 0.019374 Skewness 0.442601 0.270155 Kurtosis 6.536769 6.659862 Jarque-Bera 672.9235 729.9479 Probability 0.000000 0.000000 Sum -0.632665 -0.921083 Sum Sq. Dev. 0.548533 0.480084 Observations 1215 1280

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Figure 1 Logarithm of Oil Futures Price and Returns

Figure 2 Theoretical Quantile-Quantile

3.2 3.4 3.6 3.8 4.0 4.2 4.4 4.6 4.8

III IV I II III IV I II III IV I II III IV I II III IV I II

2011 2012 2013 2014 2015 2016

Logarithm of WTI daily oil futures prices

-.12 -.08 -.04 .00 .04 .08 .12 .16

2011 2012 2013 2014 2015 2016

WTI oil futures price returns

-.12 -.08 -.04 .00 .04 .08 .12

II III IV I II III IV I II III IV I II III IV I II III IV I II

2011 2012 2013 2014 2015 2016

Brent oil futures price returns

3.2 3.4 3.6 3.8 4.0 4.2 4.4 4.6 4.8 5.0

II III IV I II III IV I II III IV I II III IV I II III IV I II

2011 2012 2013 2014 2015 2016

Logarithm of Brent daily oil futures prices

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Table 2 Autocorrelation Statistics

WTI Brent

Lag AC PAC Q-stat AC PAC Q-stat

1 -0.106 -0.106 13.790*** -0.077 -0.077 7.6491*** 2 0.026 0.015 14.616*** 0.022 0.016 8.2703** 3 0.011 0.016 14.775*** 0.003 0.006 8.2834** 4 0.017 0.020 15.128*** 0.018 0.019 8.7172* 5 -0.039 -0.037 17.033*** -0.005 -0.002 8.7440 6 0.053 0.045 20.452*** 0.048 0.047 11.711* 7 -0.017 -0.006 20.820*** -0.024 -0.017 12.446* 8 0.035 0.032 22.299*** 0.038 0.033 14.336* 9 -0.033 -0.027 23.634*** -0.060 -0.055 18.942** 10 0.039 0.030 25.523*** 0.070 0.060 25.352*** 11 -0.022 -0.012 26.140*** -0.034 -0.023 26.827*** 12 -0.022 -0.030 26.737*** -0.046 -0.055 29.562***

Note: *, **, *** indicates rejection at the 10%, 5% and 1% significance level respectively.

The persistence of volatility is an indication of the existence of conditional variance autocorrelation (which refers to the time that needed for the effects of market shocks in crude oil futures markets to disappear). Table 2 represents the autocorrelation characteristic (autocorrelation and partial autocorrelation) for the daily returns for lags from 1 to 12 as well as Ljung-Box Statistics. The null hypothesis here is that “There is no autocorrelation.” The Ljung-Box statistic for serial correlation shows that the null hypothesis of no autocorrelation of the 1st order of WTI crude oil futures is rejected on the significance level of 1% and the no autocorrelation of the 1st order of Brent crude oil futures is also rejected on the 1% significance level. Therefore, we can confirm serial autocorrelation existence in the WTI and Brent crude oil futures returns. The autocorrelations for the crude oil futures prices returns show non-random behaviors at those lags and also indicate evidence of persistence in crude oil futures price returns. Eviews results which are not reported, also show significant positive serial correlation for short lags in the absolute and squared returns of WTI and Brent crude oil futures price returns series, indicating time dependence in the variance.

The autocorrelation characteristics of the absolute and squared returns are consistent with the definition of volatility clustering where large price changes tend to be followed by large price changes and small price changes tend to be followed by small price changes over consecutive days. Volatility clustering properties in commodity returns and its financial derivatives price returns have been examined in many previous studies using GARCH-type models (Kroner et al., 1993; Engle, 1982).

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unit root and the time series is non-stationary”. Both unit root tests for non-stationarity in the returns of WTI and Brent crude oil futures prices indicate no evidence of non-stationarity in crude oil futures price returns.

Table 3 Unit Root Tests

Time sequences

ADF PP Significance level Lag

length Bandwidth 1% 5% 10% Rwti -38.77927 -38.66540 -3.435523 -2.863712 -2.567977 0 7 D(rwti) -15.85526 -601.3783 -3.435599 -2.863746 -2.567995 16 228 D(rwti, 2) -16.47809 -713.4865 -3.435631 -2.863760 -2.568002 22 114 Rbrent -38.60962 -38.52984 -3.435251 -2.863592 -2.567912 0 7 D(rbrent) -14.67262 -655.8188 -3.435336 -2.863629 -2.567932 20 285 D(rbrent, 2) -17.39830 -1164.860 -3.435344 -2.863633 -2.567934 21 303 Note: ADF lag length automatically chosen based on SIC, maxlag= 22. PP Newey-West Bandwidth automatically chosen based on Bartlett kernel.

To test for heteroskedasticity, we use ARCH test suggested by Engle (1982), which tests the null hypothesis of “no ARCH effect” based on an ARMA (1,1) model. The ARMA (1,1) models show evidence of high statistics on squared residuals. After estimating the ARMA (1,1) model, we test ARCH effect based on 5 lags (5 trading days per week), the probability of F-statistics is 0 which suggests that we should reject the null hypothesis and there exists significant ARCH effect. This suggests that the returns on crude oil futures price suffer from heteroscedasticity. To eliminate the influence of heteroscedasticity, we follow the suggestions of Engle (1982) to model crude oil futures price returns in an ARCH/GARCH framework.

4.4 GARCH model

As Sadorsky (2006) stated, GARCH model is based on five-year rolling window. Five years of daily trading data are used to estimate the GARCH (1,1) model and a daily volatility forecast is made afterwards. According to Bollerslev and Wooldridge (1992), we selected the “Heteroskedasticity Consistent Covariance- Bollerslev & Wooldridge” option because we suspect that the residuals are not conditionally normally distributed.

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close to unity (in this case, the sum of coefficients of both WTI and Brent crude oil futures price returns are 0.996154 and 0.997027, which close enough to 1). A large sum of the two coefficients suggests that either a large positive or negative return will lead the future forecasting of the volatility to be too high for the next period. This indicates that the market shocks are persistent and the shocks to the conditional variance of crude oil futures continue overtime (the shocks don’t disappear after a short period).

Cheong Vee, Nunkoo Gonpot and Sookia (2011) suggested in their research that GARCH models with GED perform better than GARCH with student’s t distribution. Therefore, we forecast based on GARCH (1,1) model with GED and the result is shown as below. The figure of dynamic forecasts shows a completely flat forecast structure for the mean (due to the conditional mean equation includes only a constant term), which at the end of the in-sample estimation period, the conditional variance value reaches a historically low level relative to its unconditional average value. The static forecasts roll one-step ahead of the actual date, that’s the reason why the static forecast shows much more volatility than for the dynamic forecasts.

Figure 3 Dynamic forecasts of the conditional variance of WTI crude oil futures price returns based on GARCH model -.06 -.04 -.02 .00 .02 .04 .06

2012 2013 2014 2015 2016 RWTIF ± 2 S.E. Forecast: RWTIF Actual: RWTI Forecast sample: 9/02/2011 4/29/2016 Adjusted sample: 9/05/2011 4/29/2016 Included observations: 1215 Root Mean Squared Error 0.021251 Mean Absolute Error 0.014872 Mean Abs. Percent Error NA Theil Inequality Coefficient 0.993417 Bias Proportion 0.000324 Variance Proportion 0.999672 Covariance Proportion 0.000003 Theil U2 Coefficient NA Symmetric MAPE 192.9777 .00060 .00065 .00070 .00075 .00080 .00085

2012 2013 2014 2015 2016

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Figure 4 Static forecasts of the conditional variance of WTI crude oil futures price returns based on GARCH model

In the Figure 3 and 4, the conditional variance forecasts provide the basis for the standard error bands (the two red lines) and the conditional mean forecasts (the blue line). The graph in the left bottom shows that the conditional variance forecasts decrease gradually when the forecast time increases. This explains why the two standard error bands converge slightly.

4.5 EGARCH Model

GARCH models typically show a high degree of volatility persistence, for instance volatility shocks which occurred in the past continue to influence the current volatility. On the basis of the existence of high degree of volatility persistence, we assume that the conditional volatility is highly predictable. Structural breaks in the volatility process and asymmetry of the shock effects could be the reasons of volatility persistence (Diebold, 1986). It is not clear that those will influence the model estimates accuracy. Also, due to the fact that GARCH model can’t distinguish positive and negative shocks, we introduce EGARCH model to estimate the volatility.

We assume a conditional normal distribution on the time series, and specify an ARMA(p,q) term for EGARCH (1,1) model. The reasons we introduce EGARCH model are: first, there is no restriction on the parameters 𝛼, 𝛾, 𝛽 (GARCH model has restrictions on those parameters), since the variable here is the logarithm form of variance (instead of standard deviation) is modelled, then even if the parameters results are negative, the variance term will still be positive. So there is no need to put

-.12 -.08 -.04 .00 .04 .08 .12

2012 2013 2014 2015 2016 RWTIF ± 2 S.E. Forecast: RWTIF Actual: RWTI Forecast sample: 9/02/2011 4/29/2016 Adjusted sample: 9/05/2011 4/29/2016 Included observations: 1215

Root Mean Squared Error 0.021251 Mean Absolute Error 0.014872 Mean Abs. Percent Error NA Theil Inequality Coefficient 0.993417 Bias Proportion 0.000324 Variance Proportion 0.999672 Covariance Proportion 0.000003 Theil U2 Coefficient NA Symmetric MAPE 192.9777 .0000 .0004 .0008 .0012 .0016 .0020 .0024 .0028 .0032

2012 2013 2014 2015 2016

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non-negativity constraints on the three parameters. Second, Nelson (1991) mentioned that compared to the pure GARCH specification, the 𝛼 parameter in EGARCH model represents the size effect of the conditional shocks on the market on the conditional variance. Third, since the 𝛽 parameter can be either positive or negative, EGARCH model can distinguish between the positive effect and the negative effect. The 𝛽 parameter also allows us to evaluate whether the shocks from returns to volatility is persistent or not. Last but not least, asymmetries terms are allowed under the EGARCH model- if the relationship between price returns and volatility is negative, term γ in the formula will be negative (which is not the case in GARCH model). If γ >0, the formula suggests that positive shocks give rise to higher volatility than negative shocks, and if γ <0, negative shocks on the market price will influence the volatility more than the positive shocks.

We estimate the EGARCH (1,1) model as following. At first, when we estimated the model, we followed what Nelson suggested, we used a generalized error distribution structure for the error distribution. However, due to the reasons of computational ease and intuitive interpretation, we then employed conditionally normal (Gaussian) error distribution instead of generalized error distribution. The EGARCH (1,1) model results of the WTI and Brent crude oil futures price returns are shown in the Appendix 3.

Figure 5 Dynamic forecasts of the conditional variance of WTI crude oil futures price returns based on EGARCH model -.06 -.04 -.02 .00 .02 .04 .06

Root Mean Squared Error 0.021248 Mean Absolute Error 0.014877 Mean Abs. Percent Error NA Theil Inequality Coefficient 0.969756 Bias Proportion 0.000041 Variance Proportion NA Covariance Proportion NA Theil U2 Coefficient NA Symmetric MAPE 179.8032 .0003 .0004 .0005 .0006 .0007 .0008 .0009

2012 2013 2014 2015 2016

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Figure 6 Static forecasts of the conditional variance of WTI crude oil futures price returns based on EGARCH model

We used the data sample of the full sample period. For both WTI and Brent crude oil futures price returns, we noticed that coefficient on 𝛾 parameter is negative and is statistically significant at 1% significance level. Since 𝛾 measures the asymmetry of the shocks, the negative sign of the 𝛾 suggests that negative shocks on the market will reduce the volatility of the WTI and crude oil futures returns more than the positive shocks will affect. The result shows that the market shocks have asymmetric effects on the volatility of crude oil future prices. Then we look at the 𝛽 parameter, the coefficients of the both 𝛽 parameters are close enough to 1 (almost 1) and statistically significant at 1% significance level. 𝛽 parameters show the persistence of the shocks influence on volatility. The model results show that the shocks to crude oil future price returns volatility do not die rapidly but continue to influence the volatility for a long time. Since the shocks tend to persist and continuously affect the crude oil futures returns, we can conclude that the crude oil market shocks have permanent effects on crude oil futures volatility (proven by both WTI and Brent crude oil futures). We then forecast the volatility based on the EGARCH (1,1) model.

4.6 TGARCH Model

Engle (1993) suggested that in the financial markets, the downward movements of the financial markets always are always followed by higher volatilities than the same size of upward movements. One of the methods to solve this problem is to build TGARCH model.

-.10 -.05 .00 .05 .10

2012 2013 2014 2015 2016 RWTIF ± 2 S.E. Forecast: RWTIF Actual: RWTI Forecast sample: 9/02/2011 4/29/2016 Adjusted sample: 9/05/2011 4/29/2016 Included observations: 1215 Root Mean Squared Error 0.021248 Mean Absolute Error 0.014877 Mean Abs. Percent Error NA Theil Inequality Coefficient 0.969756 Bias Proportion 0.000041 Variance Proportion NA Covariance Proportion NA Theil U2 Coefficient NA Symmetric MAPE 179.8032 .0000 .0005 .0010 .0015 .0020 .0025

2012 2013 2014 2015 2016

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In the TGARCH model, we estimated the threshold term as 1. By adding the asymmetric leverage effect (using the dummy variable for past negative returns), we ensure that negative returns will generate higher level of volatility fluctuation than positive returns of the same size. The sign of the residuals does not matter in TGARCH model due to the reason of squaring. The model result of TGARCH model with threshold order of 1 and GED for WTI crude oil futures returns is shown in the appendix 6. The coefficient of the lagged squared residuals is 0.003935, which is not statistically significant at 10% significance level. The coefficient of the dummy variable is 0.084997 and the coefficient of the lagged conditional variance is 0.951502, both statistically significant at 1% significance level. The forecasting of volatility based on the TGARCH model is shown in figure 7 and figure 8. Compared to forecasting figures of GARCH (1,1) and EGARCH (1,1) model, the TGARCH forecasting on conditional variance indicates more fluctuations.

Figure 7 Dynamic forecasts of the conditional variance of WTI crude oil futures price returns based on TGARCH model -.06 -.04 -.02 .00 .02 .04 .06

Root Mean Squared Error 0.021248 Mean Absolute Error 0.014872 Mean Abs. Percent Error NA Theil Inequality Coefficient 0.979430 Bias Proportion 0.000014 Variance Proportion NA Covariance Proportion NA Theil U2 Coefficient NA Symmetric MAPE 184.1730 .0004 .0005 .0006 .0007 .0008 .0009

2012 2013 2014 2015 2016

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Figure 8 Static forecasts of the conditional variance of WTI crude oil futures price returns based on TGARCH model

5. Empirical Results and Model Comparison

We estimated GARCH, EGARCH and TGARCH models with normal, student’s t, and GED distributions with ARCH terms and GARCH terms vary from 1 to 3. Some of the model estimates are shown in the appendix. Following the model building methods of previous literatures, all inferences are based on the Heteroskedasticity Consistent Covariance Bollerslev & Wooldridge standard errors. All models have been selected based on the value of SC.

Table 4 Results from estimation of the model WTI Crude Oil Futures

𝛼 𝑡𝛼 𝛽 𝑡𝛽 SE(𝛽) 𝛾 𝑡𝛾 𝛿 𝑡𝛿 R2 AIC SC GARCH Normal 0.077 4.473 0.920 61.244 0.015 -0.000 -5.244 -5.227 T 0.072 4.895 0.925 63.084 0.0147 -0.000 -5.263 -5.242 GED 0.074 4.892 0.923 59.137 0.016 -0.000 -5.264 -5.243 EGARCH Normal 0.066 4.175 0.994 506.114 0.002 -0.077 -8.632 -0.000 -5.276 -5.256 T 0.069 3.599 0.995 458.418 0.002 -0.074 -6.360 -0.000 -5.285 -5.260 GED 0.067 3.45 0.995 443.201 0.002 -0.076 -6.573 -0.000 -5.287 -5.262 TGARCH Normal 0.002 0.154 0.951 95.534 0.010 0.091 4.648 -0.000 -5.270 -5.249 T 0.004 0.376 0.952 91.377 0.010 0.085 4.763 -0.000 -5.280 -5.255 GED 0.003 0.300 0.951 87.340 0.011 0.088 4.881 -0.000 -5.282 -5.257

Note: Results from estimation of different models, i.e. estimated parameters, t-statistics, and, for β only, standard error, R2, AIC, SC, an F-test on the unbiasedness of the model

-.10 -.05 .00 .05 .10

Root Mean Squared Error 0.021248 Mean Absolute Error 0.014872 Mean Abs. Percent Error NA Theil Inequality Coefficient 0.979430 Bias Proportion 0.000014 Variance Proportion NA Covariance Proportion NA Theil U2 Coefficient NA Symmetric MAPE 184.1730 .0000 .0004 .0008 .0012 .0016 .0020

2012 2013 2014 2015 2016

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We can derive several conclusions from the GARCH models. The first conclusion is that, according to the information criteria (Akaike info criterion and Schwarz criterion), complicated GARCH type models with higher order of ARCH term and GARCH term don’t provide better estimates (evidence to support hypothesis 2). The estimate results obtained from GARCH (1,1) model don’t seem to be improved. This conclusion is proven by the literature from Fong and See (2002) and Agnolucci (2009), which also used GARCH(1,1) model for forecasting WTI crude oil futures volatility. Javed and Mantalos (2013) explained in their finding that “performance of the GARCH (1,1) model is satisfactory” due to the first lag of conditional volatility is sufficient enough to capture the movements of the conditional volatility. Secondly, the choice of different error distributions does not affect the value of the estimated coefficients. All coefficients on the GARCH terms vary from 0.92 to 0.99 (almost the same for different distributions among the same model, for instance, 0.92 for GARCH(1,1) model, 0.99 for EGARCH (1,1) model and 0.99 for TGARCH model), while those coefficients on ARCH terms also fall in the similar ranges. This finding has been proven by Agnolucci (2009), Hentschel (1995) Fong and See (2002). The sum of the parameters coefficients which influence the persistency of the shocks, 𝛽, are above 0.99, close enough to 1, indicates that the market shocks to those time series data are highly persistent, and won’t disappear with time. Thirdly, normality of residuals has been proven to be a problem in estimate models. The normal Gaussian distribution model performs worse than the GED and student’s t distribution (Fong and See, 2002). But adding asymmetric terms like EGARCH and TGARCH models does not add much value to the explanatory ability of the models.

Results on the Bayesian information criterions are shown in the table 4. When selecting models, academics prefer to use Schwarz criterion rather than the Akaike information criterion. The smaller Schwarz criterion suggests a better fitted model. In table 4, both AIC and SC suggest that EGARCH models perform better than GARCH (1,1) and TGARCH model. They also confirm that GARCH-type models with generalized error distribution have the best performance. We then accept the hypothesis 3 that GED in GARCH models provide better performance than the others. However, concerning the question that Agnolucci (2009) suggested “whether the comparison of volatility forecasting models is influenced by the criterion used in the exercise”, the answer is no. Compared to the findings of Wei, Wang and Huang (2010), that the criterion affects the comparison of the volatility models, AIC, SC, RMSE, MAE, MAPE, SMAPE and Theil U2 all prove that EGARCH (1,1) model with student’s t distribution is the best fitted model for WTI and Brent crude oil futures.

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(MAD) and the Theil U statistic. These forecast summary statistics are important for comparing different models but they can’t provide test statistics of the difference between different models (Diebold, 1998).

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Figure 9 Forecast comparison graph based on WTI futures

Table 5 Forecasting Statistics for WTI Crude Oil Futures for time period 1/1/2016-4/29/2016

RMSE MAE MAPE SMAPE Theil U1 Theil U2

GARCH Normal 0.000673 0.000540 32.73706 35.26028 0.204737 3.665887 T 0.000682 0.000553 33.66569 36.01695 0.207679 3.742119 GED 0.000677 0.000546 33.16353 35.59452 0.206033 3.700194 EGARCH Normal 0.000630 0.000503 27.78061 33.45746 0.217288 2.983848 T 0.000608 0.000474 25.74334 30.75991 0.207398 2.832525 GED 0.000617 0.000486 26.65124 31.95106 0.211314 2.897139 TGARCH Normal 0.000634 0.000502 27.84441 33.68264 0.218469 3.018821 T 0.000631 0.000490 26.76290 32.29519 0.216136 2.952772 GED 0.000632 0.000495 27.26149 32.94159 0.217078 2.984519 IV 0.000673 0.000540 44.79212 35.26028 0.204747 4.806927 .000 .001 .002 .003 .004 21 28 4 11 18 25 1 8 15 22 29 7 14 21 28 4 11 18 25 M12 M1 M2 M3 M4

IVV GARCHNORMAL GARCHT GARCHGED EGARCHNORMAL EGARCHT EGARCHGED TGARCHNORMAL TGARCHT TGARCHGED IVV

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6. Conclusion

We can draw several conclusions from the research. From the perspective of volatility model estimation, at first, the mean returns from the crude oil futures are negative and close to 0, though the mean returns are always statistically insignificant. Secondly, the shock persistency parameters coefficients are close to 1, so the market shocks to the conditional variance of crude oil futures are highly persistent. Thirdly, the influence of the error distributions is not significant, since all the coefficients of ARCH and GARCH terms are similar. In addition, leverage effect can be observed from the crude oil futures, since the EGARCH models and TGARCH models perform better than the GARCH (1,1) model. The negative shocks have more volatility responses than the same magnitude positive shocks. Based on the data from 2011 to 2016, we would suggest EGARCH (1,1) with GED is the best fitted model for WTI and Brent crude oil futures market.

From the perspective of forecasting accuracy, GARCH-type models perform better than the implied volatility model (obtained from the Black-Scholes equation). Among the GARCH-type models, EGARCH model has the best forecasting ability. In WTI and Brent crude oil futures markets, the negative shocks influence more on conditional volatility than the positive shocks of the same magnitude. And among the same type of GARCH models, student’s t distribution generates better performance than normal Gaussian distribution and generalized error distribution. However, even though implied volatility model performs worse than all GARCH type models, there is still some information contained in implied volatility model that cannot be obtained from GARCH type models.

There are two main implications from this thesis. Firstly, when we consider the full sample from 2011 to 2016, the evidence shows that the crude oil futures market has asymmetric information and the different movements of shocks lead to asymmetric effects on market price and returns. A negative shock that increases the crude oil futures price cannot be completely redeemed for by a positive shock that decreases the crude oil futures price. The asymmetry effects indicate that crude oil futures prices might suffer the effects from regime shifts. It will be an interesting improvement to this study to add Markov regime-switching models into estimating volatility. Secondly, the crude oil futures price is not stable, which indicates high level of volatility. There are so many factors affecting the crude oil futures prices in different markets, such as the supply demand relationship, economic cycle, policy, season, society, psychological behaviors and so on. Thus, market shocks, especially political and economic shocks, for instance, new policy on environment protection, new import and export regulation, will lead to high fluctuation of the crude oil futures price.

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market shocks, for instance, the 2008 global financial crisis, to generate a general estimation model or try to distinguish the low volatility period with the high volatility period. In summary, energy economists, academics, government policy makers and investors should not rely on only one volatility estimation and forecasting model to make decisions. The accuracy of different models depends on the different data sample, forecasting purpose, risk attitude and loss function. It would be better to combine current existing research to gain a general idea of the energy market.

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Comparable performance of GARCH-type models and implied volatility model on energy futures volatility estimation: Using crude oil futures as an example

Master Thesis