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Diploma Thesis, 2010
102 Pages, Grade: 1,3
1 Introduction
2 Volatility: Concept and Empirical Evidence
2.1 Definition of Price Volatility
2.2 Analysis of Oil Price Volatility
2.3 Economic Sources of Price Volatility
3 Methodology
3.1 Modelling Oil Price Volatility
3.2 Univariate Volatility Models
3.2.1 Historical Volatility
3.2.2 ARCH
3.2.3 GARCH
3.2.4 GJR-GARCH
3.2.5 ARMA-GJR-GARCH
3.2.6 Conditional Distribution
3.2.7 Regime-Switching Models
3.2.8 Markov-Switching Models
3.2.9 Markov-Switching ARCH (SWARCH)
3.2.10 Indicators of synchronization
3.3 Bivariate GARCH Models - Volatility Transmission Models
3.3.1 VECH
3.3.2 BEKK GARCH (1,1)
4 Financial Market and Oil Price Volatility: Contagion and Transmission Channels
4.1 Definition of Contagion
5 Empirical Section
5.1 Data description
5.2 Estimation of Oil Price Volatility
5.2.1 Univariate GARCH (1,1) and GJR-GARCH (1,1)
5.2.2 Results of GARCH (1,1) and GJR-GARCH (1,1) Models
5.3 Volatility Transmission
5.3.1 Markov-Switching Estimation
5.3.2 Indicator of Synchronization
5.4 Estimation of BEKK GARCH (1,1)
5.4.1 Testing for Volatility Transmission
5.4.2 Empirical Results
6 Discussion of the Empirical Results
6.1 Volatility Transmission Channel – Financial Deleveraging
6.2 Extension of the model
6.2.1 Overvaluation and GARCH (1,1)
7 Conclusion
8 References
Hiermit versichere ich, dass ich die vorliegende Diplomarbeit selbständig verfasst und keine anderen als die angegebenen Quellen und Hilfsmittel verwendet habe. Ferner versichere ich, dass die Arbeit nicht an anderer Stelle, auch nicht teilweise, eingereicht wurde und dass ich alle Stellen, die wörtlich oder sinngemäß aus anderen Quellen entnommen wurden, als solche kenntlich gemacht habe.
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Besonders in jüngster Zeit kommt der Analyse von Ölpreisvolatilität aus volkswirtschaftlicher Sicht eine bedeutende Rolle zu. Gegenwärtig werden bestimmte Rohstoffe wie Rohöl als relevante Anlageinstrumenten von Investoren benutzt, um sich gegen Risiken an den Finanzmärkten abzusichern.
Diese Diplomarbeit beschäftigt sich mit der Berechnung von Ölpreisvolatilität in der Zeitperiode von Januar 2002 bis Juli 2009. Dabei werden Berechnungen von Ölpreisvolatilität während der Finanzkrise im Jahre 2008 untersucht. Diese Finanzkrise hat sich tiefgreifend auf die Entwicklung der Preise von Kapital- und Finanzgütern ausgewirkt. Dabei weisen die exzessiven gemessenen Werte von Preisvolatilität während der Finanzkrise auf eine strukturelle Veränderung der Preisbildung von Kapital- und Finanzgütern an den Kapital- und Finanzmärkten hin. Interessanterweise lassen sich bei der Analyse von Ölpreisvolatilität bedeutende Fakten feststellen, deren Existenz die gegenwärtig verwendeten statistischen Modelle, die sich mit der Messung von Preisvolatilität befassen, in künftigen Arbeiten komplementieren könnten.
Im Rahmen dieser Diplomarbeit werden fünf wichtige statistische Modelle analysiert: ARCH, GARCH, BEKK-GARCH und Markov-switching Modell. Dazu wird aus den Ölpreisdaten der letzten 8 Jahre die tägliche Preisvolatilität berechnet, um mögliche Relationen zwischen der Volatilität am Ölmarkt und der Volatilität am Finanzmarkt zu untersuchen. Dabei werden diese implementierten Verfahren auf ihre Gültigkeit in Berechnung und Vorhersage von plötzlichen Preisveränderungen untersucht. Insbesondere wird darauf eingegangen unter welchen Bedingungen die Verfahrensergebnisse als zuverlässig gelten.
Diese Diplomarbeit wurde im Rahmen eines Forschungspraktikums bei der Organisation erdölexportierender Länder (OPEC) in Wien, Österreich unter Betreuung des Lehrstuhls für Wirtschaftstheorie der Universität Potsdam, fertiggestellt.
Figure 1. Logarithm of daily oil prices (WTI). Obs. 1/2/2002 – 7/8/2009.
Figure 2. Returns in (a) oil price and (b) S&P 500 Index. Obs. 1/2/2002 – 7/8/2009.
Figure 3. Autocorrelations of oil price returns residuals with different lags with 5% significance.
Figure 4. WTI crude oil distribution of daily returns. Obs.1/2/2002 – 7/8/2009.
Figure 5. Logarithm of daily prices and volatility for the WTI crude oil.
Figure 6. Daily oil price and oil price volatility (conditional variance). Obs. 6/6/2008 – 6/6/2009.
Figure 7. Conditional variance of Standard and Poor’s 500 Index and WTI crude oil.
Figure 8. Standard and Poor’s 500: Daily index.
Figure 9. S&P 500 Index daily returns and bivariate SWARCH(2,1) smoothed probabilities of volatility regimes.
Figure 10. WTI crude oil: Daily spot price.
Figure 11. WTI crude oil daily returns and bivariate SWARCH(2,1) smoothed probabilities of volatility regimes.
Figure 12. WTI Crude and S&P 500 Index bivariate SWARCH(2,1) smoothed probabilities of volatility regimes.
Figure 13. WTI Crude and S&P 500 Index bivariate SWARCH(2,1) smoothed probabilities of volatility regimes. Obs. 8/8/08 – 9/25/08.
Figure 14. Synchronization.
Figure 15. (De-)Synchronization between financial market and oil market.
Figure 16. Conditional covariance (BEKK GARCH) between Standard and Poor’s 500 Index and WTI crude oil market returns.
Table 1. ADF test for stationarity.
Table 2. Autocorrelations of WTI squared residuals.
Table 3. Descriptive statistics for oil price returns.
Table 4. GARCH(1,1) model results for WTI Crude Oil.
Table 5. GARCH(1,1) for S&P 500 Index.
Table 6. GJR-GARCH(1,1) model results for WTI Crude Oil.
Table 7. GJR-GARCH for S&P 500 Index.
Table 8. Hamilton's Markov-switching model with two regimes.
Table 9. Markov transition probabilities.
Table 10. Indicators of (de-) synchronization.
Table 11. BEKK MGARCH(1,1). Obs. 1/4/2002 – 6/10/2008.
Table 12. BEKK MGARCH(1,1) . Obs. 6/11/2008 - 6/25/2009.
The financial distress of 2008 had a significant impact on the market structure of financial assets causing high levels of volatility.^{[1]} During the last two decades a new concept has had a huge influence on economic thinking, namely volatility transmission. Thus, this represents a new issue of economic analysis and has attracted the attention of many economists, econometricians and policy makers. The idea that asset returns fluctuations in one market spill over into another is not new, and has been analysed profoundly by many researchers such as Engle, Ewing and Forbes. The increasing financialization of commodities and expanding OTC derivatives markets indicate that the interrelation between global financial markets and commodity markets – in this case oil – has increased as well.^{[2]} Several authors have argued that the deregulation of derivatives markets could cause significant fluctuation of commodities and other financial assets. Unstable financial markets can be still exacerbated by excessive amounts of uncovered collateral debt at OTC markets, which lack of a clearing house, jepardizing the relationship between dealers. The possibility of volatility transmission between two markets lies at the heart of the problem of spreading out of financial or economic shocks around the globe.
Figure 7 in section 4 shows the WTI crude oil. This graph indicates that during periods of financial turmoil oil price returns seem to present an increased volatility. Therefore, volatity has been present in both markets simultaneously, so that it appears to have been originated in one market and then spread over to other previously unaffected markets. According to the Dodd-Frank Wallstreet Act the OTC markets must be regulated after the evidence of excesive risk that was taken by market participants and brokers who could not hold bilateral forward contracts as cause of partial or total loss of the necesary collateral amount to commit the contracts, leading to deleravaging. The leveraged amount of collateral shows the financial links between both markets. Many theories can be applied to explain the phenomena of the significant high persistent volatility in the WTI crude oil market during the financial crisis. The cognition of the persistence of volatility permits to obtain more efficient parameter estimates, as persistence suggests that current volatility can be predicted.
In order to analyse the persistence of volatility I will apply, on the one hand, univariate volatility models that are able to estimate the volatility persistence of the time series of each market. These econometric models are the autoregressive conditional heteroscedasticity (ARCH) model developed by Engle (1982), and the generalized autoregressive conditional heteroscedasticity (GARCH) by Bollerslev (1986). These two models are by far the most popular method used for analysing high-frequency financial time series data. The causes of the spillover effect on the markets are generally assigned to cross-market hedging and changes in usual information, which may modify expectations across markets. On the other hand, I will apply multivariate volatility models to indicate the volatility transmission channels between the stock market and the oil market. The comparison of the resulting parameters generated by the univariate and multivariate volatility models has a principal relevance to measure the volatility transmission magnitude.
In order to analyse the events of financial stress, it is necessary to apply the Markov-switching techniques. This model is particularly well suited to differentiate among different volatility regimes (e.g., low and high), deduced from time-varying nature of volatility that occurs in many high-frequency financial variables, especially during times of financial distress.^{[3]} This econometric method allows for a time-varying variance and regime-dependent ARCH parameters exhibited in the financial data under spinning the oil market. Moreover, this technique permits to describe the “system’s dynamics” to switch between two states. This model is broadly used to carry out analysis concerned to the status of “systemic risk” when certain variables (e.g., bank stability indicators or global market variables), change their volatility (or mean) states.^{[4]} The behaviour of the market participants may change dramatically in uncertain periods, causing a structural break in a series.
In this paper, the author uses data of daily WTI crude oil and S&P 500 to explore more closely the relationship of the volatility between the financial and the oil market. A research takes place as to whether periods of increased stock market volatility coincide with the financial crises of 2008 within the sample period. Inquiring deeply into these economic phenomena would be highly valuable and could have important implications for policy and regulatory decisions. Any evidence for a possible transmission of volatility across markets would favour arguments to regulate the OTC derivatives markets through the implementation of a central clearing house which requires from markets participants the necesary margins to hold bilateral contracts until the maturity date. By doing this, the participation of non-commercial derivative market participants would show more transparency.
This approach is in line with other studies analysing the effects of financial crises “on assets” volatilities across countries. Examples of papers on this subject include Bennett and Kelleher (1988), King and Wadhwani (1990), Engle, Ito, and Lin (1990, 1992) and Hamano, Ng, and Masulis (1990). Other papers dealing with possible spill-over of financial markets’ volatility are Edward and Susmel (2000, 2001) investigating interest rate volatilities and stock market returns in emerging markets respectively. These features of the nature of oil prices have been analysed by many financial researchers. In this work I will mention some of the most influential researches that have contributed to the oil price analysis.cNowadays the examination of the factors which induce oil price volatility should reach beyond mere supply and demand for the commodity.^{[5]}
This study is based on the previous research by moving from the framework of commodity markets to financial linkages between financial and physical markets. Thus this work aims to observe and explain the volatility spillover from the financial market volatility to the oil market. I will apply multivariate GARCH to estimate simultaneously the means and conditional variances of the Standard & Poor 500 Compound Index (S&P 500) and West Texas Intermediate (WTI) crude price returns to analyse its transmission.^{[6]} To achieve the estimation of volatility transmission or spillover between the financial markets and oil market I apply the BEKK representation of a multivariate GARCH (p,q) process model proposed by Engle and Kroner (1995), which does not restrict the of constant conditional correlations across and between the financial market and the oil market. This econometric time series model permits an examination of covariance spillovers across several markets as well as a calculation of hedging ratios.
The objective of this work is to analyse the structure of volatility parameterization of two financial time series: oil market and financial market, for a given time period. Furthermore, the examination of their returns and of the interaction between their returns should demonstrate whether they are related to each other. The following questions are in the scope of this research: (a) Is there empirical evidence that the oil market volatility was impacted by the financial market? (b) What is the role that the time plays in the formation and transmission of volatility? (c) What are the factors that cause significant levels of volatility / variance in the asset price formation? (d) Are the usual econometric models capable to capture persistent volatility periods? (e) Are these models able to forecast short-term oil price movements? The fact that the varying trading volume can affect the financial assets pricing in different markets has already been researched by Fleming and Ostdiek (1998). They attempt to demonstrate whether the introduction of crude oil futures and subsequent introduction of energy derivatives exacerbated oil price volatility. Their findings attest that large unexpected abnormal volatility occurred after the introduction of the futures contracts. This factor is actually relevant for the research on the relation between oil price volatility and the information availability of the oil futures markets.
This thesis is organized as follows. Section 1 is the introduction. Section 2 focuses on the concept of volatilty and empirical evidence of it. Further, different possible transmission channels across countries for financial crises are presented and there is an overview of the most recent existing empirical literature analysing contagion. In Section 3 the applied econometric methodology along the thesis is presented and explained. This attempts to comprehend and to measure the existence of spillover effects of volatility from one market to another resulted from the increasing integration of the global financial markets.^{[7]} Section 4 reports the results obtained and Section 5 concludes the study.
The software programs used for the econometric estimations of this thesis are STATA 10, Ox Metrics 6.0 and Microsoft Excel 2007.
Price volatility is defined as a measure of the dispersion in a probability density of rate returns of a time series^{[8]} and denotes the degree to which the price of an underlying asset tends to vary over time. To estimate the dispersion of a population the most common measure is the standard deviation σ of a random or stochastic variable, thus, the square root of its variance.^{[9]} Volatility is the annualized standard deviation of price returns. Assuming that the time periods between the price process observations converge to zero; the approximation of volatility is a function of the price returns, rt, squared:
illustration not visible in this excerpt (2.1)
The estimation of historical volatilities is related to the measure of cointegration between two assets. The degree of correlation between daily financial returns and daily oil price returns is a measure of comovements between these two return series. A strong positive correlation indicates that upward fluctuations of one return series tend to be accompanied by upward movements in the other, and similarly downward movements of the two series tend to go together. Contrarily, a strong negative correlation shows upward fluctuations in one series and downward movements in the other.
Unlike prices, it is not possible to observe directly volatility and correlation in the market. They can only be estimated with the application of a model.
The comprehension of oil price volatility is an important input for the economic analysis, because unexpected changes of volatility can induce persistent disequilibria periods within interrelated markets. For example, an oil price shocks could expose producers and consumers to high levels of risk, affecting investments in oil inventories and facilities for production and transportation (Pindyck, 2004). Strong oil price fluctuations can also affect market variables by influencing the total marginal cost of storage, and by influencing the relative prices of the capital structure of companies. This study aims at the determination of the nature of the high levels of oil price volatility in the last recent years. Pindyck indicates that the inclusion of volatility as a market variable can lead to a better comprehension of the short-run commodity market dynamics (Pindyck, 2004). Therefore Pindyck recommends the application of econometric methods which model the market price structure in order to analyse the oil price fluctuations of a given time series.
Following the logic of fundamental factors the oil price volatility is related to the significant low price and demand elasticity, being reflected on the changes of demand and production levels at a time period. However, the price movements of international oil markets may as well be impacted by the actions of speculators which attempt to exploit the price differentials of a commodity in two different markets to obtain profits. The speculative attacks in one specific market may cause sudden fluctuations of commodity prices, leading in extreme cases to financial panic, overpricing and herd behaviour of the market participants.^{[10]} Thus, volatility induced by speculative attacks attracts more traders and investor into the commodity market, increasing the volatility when the resulting liquidity becomes excessive. The speculators aim on the expected return which results from changing prices.
In this paper I will concentrate on the identification of volatility spillover or transmission between the financial market and the oil market in periods of financial instability.
In this study I will focus on the financial factor of the stochastic market dynamics, which according to the International Monetary Fund caused an increase of the oil price volatility in the time period between 2007 and 2008.^{[11]} The fact of the emergence of oil and commodities as an asset class indicates that the value of oil and commodities can no longer be considered as merely an industrial commodity but also as a financial asset.^{[12]} Volatility of oil price can be understood as a function of the following factors that are related to the financial markets:
a) Impact of the global interest rate levels like the London Interest Borrowing Offered Rate (e.g. LIBOR) on the storage levels, and therefore to the distortions of the supply of crude and expectations of scarce supply and rising prices. This physical factor that is determined by fundamentals generates a subjective marginal signal that causes the formation of expectations in space and time that can cause a real effect on the market. This is why I will outline the importance of knowledge and time in the development of the market structure.
b) Structure of financial markets and derivatives markets in particular. Divergence and convergence between the yield curve of financial markets and the yield curve of crude oil futures. The existing volatility is a consequence of the variance of return in the equity market.
c) Determinants for transactions on futures markets that are not related to the physical oil situation. This results as consequence of the demand and supply of oil futures contracts, which are held by market agents as financial instruments in a portfolio. This diversification of monetary wealth aims to optimise the composition of the portfolio to protect it from unexpected price fluctuations. Funds flow in and out of financial market, in form oil futures, sovereign and corporate bonds, stocks, foreign exchange, etc., depending on relative expectations (i.e. if oil prices are expected to rise by 5 per cent in the next three months but bond prices to rise by 10 percent in that same period, some funds will move from oil to bonds). The expectations about future positive oil price returns should induce attempts to buy oil contracts but if the prospects are rosier in some other market some traders will sell them off bringing the price down at least temporarily. The reverse will also occur if funds are induced to flow from a market affected by bearish expectations into oil where the prospects may not be brilliant but only less bearish than elsewhere.^{[13]}
d) Another important source focuses on the correlation between oil prices and GDP, particularly, in the emerging countries, and the potential supply shortfalls due to a higher rate of increase in demand in the product markets and the composition of supply (Jalali-Naini, 2009).
e) Supply, demand, exchange rates, inventory levels, environmental aspects (tax structure), alternative energy and innovation.
Considering the growing integration of global financial markets, it is of interest to analyse the volatility spillover effects from one market to another. Particularly Malik and Ewing (2009) suggest the existence of two fundamental transmission channels of possible volatility spillover between the financial markets and the oil markets: (1) Volatility spillovers may be transmitted through the cross-market hedging and changes in common information, which may at the same time modify expectations across markets.^{[14]} (2) Financial contagion, specifically, a shock to one country's asset market may cause changes in asset prices in another country's financial market (Malik and Ewing, 2009). Thus, investors spread out shocks among markets through the channel of cross-market balancing, by adjusting their portfolio's exposure to macroeconomic risks. If two markets are closely linked through economic fundamentals, then a distress in the first market would be expected to have a major real impact on the second market. The sources of financial contagion across markets could also be understood as consequence of certain market sensitivities of shared macroeconomic risk factors and the amount of information asymmetry among markets.
There are many different concepts that define contagion indicating the spillover process of financial distresses. The most of the contagion concepts are based on financial sectors. Due to the fact that crude oil is actually considered as a financial asset, I intend to demonstrate whether the volatility transmission from financial market to the oil market can be considered a contagion. According to the most stated definition, contagion is the cross-country transmission of shocks or the general cross-country spillover effects. This concept of contagion does not need per se to be related to crises. However, contagion has been emphasized during crisis times. This definition of contagion can be adapted on volatility transmission between two or more markets excluding geographical and national elements.
Thus, Rigobon (Forbes and Rigobon, 2000) presents three different economic definitions of contagion based on financial distresses across countries.
1. It might be interpreted as the occurrence of crisis: a collapse in one country generates a speculative attack in another one.
2. Based on the fact that countries in crisis experience increases in volatility or their returns, contagion can be characterized as the transmission of volatility across countries.
3. Contagion can be defined as a change in the propagation of shocks across countries. I.e. shocks from one country to another are transmitted with higher intensity during crisis or a significant increase in cross-market linkages after a shock to an individual country, which might be called “shift contagion” (Forbes and Rigobon, 2000).
These definitions denote the importance of the significant increase of conditional variance in one country’s market and its transmission to other markets. Whereby, this issue will be of central relevance in the following section of this paper.
In the particular case of possible volatility spillover effects from financial markets to oil markets, it is convenient to indicate that these markets present an important integration, meaning that their returns are closely related to the mechanism of hedging, which intends to protect investors from financial and macroeconomic risks. This suggests that the cross-market linkages between both markets are persistent over time. More specifically, the scope of this paper allows the adaptation of Forbes and Rigobon’s definition of contagion to the oil market, because contagion is interpreted as a significant increase in cross-market linkages after a shock to one market (or group of markets, represented by indexes). Forbes and Rigobon (2000) define this kind of contagion as a shift-contagion, indicating that the sudden increase of volatility in one market simultaneously affects, or with a given time lag, another markets returns by increasing the conditional variance of the second market that experiences a shift of the values of correlation and conditional variance to a higher level.
According to this definition of contagion, one principal question arises: should a strong increases in the first market and an increase the volatility in the second market should be labelled contagion or not? This discussion implies that if two markets are highly correlated after a shock, this is not necessarily a contagion. It can only be defined as a shift-contagion if the correlation between the two markets increases significantly. If the transmission of a large exogenous shock from one market to the other implies only the continuation of the existence of the same cross-market linkages, which are present during more tranquil periods, then this should not be considered contagion. One market can become contagioned by another market when the magnitude of the parameters related to cross-market linkages increase significantly.^{[15]} By defining contagion as a significant increase in cross-market linkages, this paper avoids a direct measure and differentiation between these various propagation mechanisms.
How are shocks transmitted across markets? In this work, we analyse the financial econometric methods to identify volatility transmission. The explanation of propagation of shocks is not in the scope of this research. This issue could be extended.
The modelling of oil price volatility has a relevant function for analysing the measure of financial uncertainty and asset pricing but also for the estimation of risk compensation, price volatility of an asset denotes an important determinant of its expected and current price. Since Mandelbrot (1963a, 1963b) and Fama (1965), it has been a well-known fact that financial asset return is not constant over time. Furthermore, there is empirical evidence that demonstrate the existence of volatility clustering in financial time series. The volatility term, illustration not visible in this excerpt of the commodity price returns is a significant parameter in financial markets that influences the decision and behaviour of the market participants. The magnitude of volatility denotes uncertainty about the returns provided by the underlying asset value and appears as reaction of prices to news (information) and shocks. Generally two types of oil price volatilities are estimated: the implied volatility^{[16]} from crude oil call options^{[17]}, and volatility computed by the standard GARCH (1,1) model. In financial computing, two categories of stochastic processes are applied to model stochastic first moments: the stochastic volatility (SV) models and the ARCH/GARCH models. Latter will be fitted to daily financial data to estimate the degrees of persistence of time series. The price volatility model GARCH (1,1) is broadly applied in financial statistics due to the analysis and estimation of the historical oil price returns as well as for forecast of next period’s returns fluctuations which denotes the presence of conditional variance and a time varying volatility.^{[18]}
Traditional time series generally assume a homoscedastic model. Moreover, in economic or financial time series, it is more realistic to presume that the conditional variance or volatility function is time-varying.^{[19]} Many macroeconomic variables and financial time series exhibit volatility clusters and tranquillity periods. During months of more news, uncertainty and trading, the asset prices move more rapidly than in tranquil months. The occurrence of volatility clusters suggests that the rate of price change in discrete time is not distributed normally, i.e. the homoscedasticity assumption is not sufficient, because the conditional variance of asset price returns is not constant. Thus, it is convenient to assume an autoregressive model for volatility,[ illustration not visible in this excerpt ](in the multivariate case, the conditional covariance matrix[ illustration not visible in this excerpt ]), in which the conditional variance is a function of previous variables only.
To demonstrate the unsatisfactory nature of standard econometric models for estimating risk and uncertainty, consider the following first order autoregressive model AR(1):
illustration not visible in this excerpt (3.1)
Considering that[ illustration not visible in this excerpt ]for condition of stationarity. In case the model intends to forecast[ illustration not visible in this excerpt ], then the unconditional mean forecast of[ illustration not visible in this excerpt ]is[ illustration not visible in this excerpt ]and the unconditional forecast error variance is[ illustration not visible in this excerpt ]In the case of conditional estimation, the conditional forecast of[ illustration not visible in this excerpt ]is[ illustration not visible in this excerpt ]and the conditional error variance is[ illustration not visible in this excerpt ]This implies the difference between unconditional and conditional forecast error variances, unless[ illustration not visible in this excerpt ], but they are both constant. One major disadvantage is that the unconditional forecast do not depend on the available information set[ illustration not visible in this excerpt ]and hence do not change over time, meaning that the conditional volatility models are more adaptable to the empirical financial time series. This implies that since illustration not visible in this excerpt , the unconditional forecast has a greater variance than the conditional forecast, which connotes that the conditional forecasts are preferable, because they take into account the known current and past realizations of series.
The returns on oil prices exhibit very high levels of frequency that may show signs of autocorrelation, so that they are not independent.^{[20]} Daily ex-post volatility (conditional variance) is measured by squared daily returns (Brailsford and Faff, 1996).
illustration not visible in this excerpt
Defining the volatility,[ illustration not visible in this excerpt ], of the annualized variance of the market variable on a time period (in this case a day)[ illustration not visible in this excerpt ]as estimated at the end of day[ illustration not visible in this excerpt ]. The square of volatility,[ illustration not visible in this excerpt ], on the time period t indicates the variance rate.
Volatility models are capable to indicate on volatility clustering during increasing uncertainty of expected oil prices returns. Contrarily, oil price volatility declines during relatively tranquil periods (or in other word, when the expectations of the market participants are being realized).
The estimation of historical volatilities is computed based on the historical underlying market price.^{[21]} Thus, the estimation model assumes that[ illustration not visible in this excerpt ]is the innovation (news) mean for energy price returns. An n -day historic volatility is on equally weighted averages of n squared returns. Based on the historical mean model, the optimal forecast for the[ illustration not visible in this excerpt ]period is the average of the previous volatilities.^{[22]} This approach suggests a stationary volatility series
illustration not visible in this excerpt (3.2)
This estimation is calculated over the last n days, where n is the amount of observations in a time period. The historical volatility assumes that volatility is constant over the estimation period and the forecast period.
The current n -day historic volatility estimate is sometimes used for the forecast of future volatility to plug into the pricing model for an option that maturates in n days. Due to the assumption of constant price volatility of the underlying asset across the estimation period, it seems not to be optimal for the estimation of oil price.
Autoregressive Conditional Heteroscedasticity (ARCH) model was originally created by Engle (1982) to analyse U.K. inflationary uncertainty. Indeed, the ARCH models have been found wide application in statistical techniques in the estimation of time varying financial volatility.^{[23]} The ARCH process interprets the conditional heteroscedasticity of financial returns by assuming that current conditional variance is a function, or more precisely a weighted average of past squared unexpected returns. The ARCH (q) model for a white noise series rt is defined by
illustration not visible in this excerpt (3.3)
Thus,[ illustration not visible in this excerpt ]denotes the conditional variance of the series, and[ illustration not visible in this excerpt ]is the available information set at time illustration not visible in this excerpt . The conditional variance in the ARCH (q) model is a function of the magnitude of previous unanticipated innovations,[ illustration not visible in this excerpt ]. Based on the assumption that variances are non-negative, the ARCH model imposes the constraints that[ illustration not visible in this excerpt ]and[ illustration not visible in this excerpt ], then the conditional variances are positively related to the value of[ illustration not visible in this excerpt ]is larger for larger values of past innovation[ illustration not visible in this excerpt ]. These features permits the prediction of time series conditional variance, when the dispersion of the residual returns is normal distributed.^{[24]} For example, if an appreciable market movement occurred yesterday, the day before or up to n days ago, the effect will be an increase in today’s conditional variance due to the fact that all parameters are constrained to be non-negative.^{[25]} However the use of ARCH model for the estimation of financial volatility is not recommended, because it requires many lags to approach the estimation to a GARCH model, which will be presented in the following section. As the lag increases in ARCH model framework it becomes more complex to estimate parameters because the likelihood function becomes flat.^{[26]}
GARCH models of conditional variance can be interpreted as an ARMA process in the squared innovations. These models have found broad application in the econometric volatility estimation of financial time series and were among the first models to take into account the volatility clustering phenomenon. These research works can be found, among other, in Bollerslev et al. (1992), Diebold and Lopez, (1995) and Poon and Granger, (2003). In order to estimate the returns of oil price, which experience heteroscedasticity, it is convenient to apply the econometric methods of Engle (1982) and Bollerslev (1986) which model the oil price return employing an ARCH/GARCH framework. When modelling and estimating the volatility term structure of energy commodity prices and financial assets it is recommended by authors to employ the univariate GARCH methodology with an extension of exogenous variables to inquire into the reciprocal relationship between previous conditional variance and current variance for individual time series. Hamao et al. (1990) carried out the first research that applied the univariate GARCH to analyse relationships between financial and derivatives markets. The Autoregressive Conditional Heteroscedasticity (ARCH) model is a predominant statistical technique employed in the analysis of time varying volatility. However this model has a severe weakness consisting that the model assumes that positive and negative shocks have the same effects on volatility because it depends on the square of the previous shocks. The univariate GARCH models denote a fundamental advantage for modelling volatility, because they generate mean-reverting term structures forecast for volatility in a simple analytic form.^{[27]}
The measure and analysis of the oil price process presents a time-varying volatility that is governed by a GARCH process, and then the realized volatility is the GARCH volatility. Bollerlev’s Generalized Autogressive Conditional Heteroscedasticity GARCH (p,q) specification (1986) generalizes the model by permitting the current conditional variance to be a time series function of the first (p) past conditional variances[ illustration not visible in this excerpt ]as well as of the (q) past squared innovations[ illustration not visible in this excerpt ]. Adrangi et al. (2001) apply GARCH to analyse the nature of non-linearities in energy prices.^{[28]} Adrangi et al. (2001) conclude that the non-linear dynamics of energy price changes can be modelled with GARCH (1,1).
Let[ illustration not visible in this excerpt ]indicate the continuously compound rate of log return of WTI crude from time[ illustration not visible in this excerpt ] , where [ illustration not visible in this excerpt ] is the crude oil price at time[ illustration not visible in this excerpt ].
The basic ARCH used for the determination of the oil price volatility consists of two equations. The mean equation analyses the behaviour of the mean of the time series; this refers to a linear regression function that includes a constant and possibly some explanatory variables. The mean function contains only an intercept.
illustration not visible in this excerpt (3.4)
The principal idea of the ARCH specification is that the variance[ illustration not visible in this excerpt ]in Equation (3.4) is higher in periods of high volatility and lower in periods of lower volatility. Thus a measure of persistence in the variance exists. Let[ illustration not visible in this excerpt ]be a white noise with unit variance. In a GARCH (p,q) model the volatility is a function on last period’s volatility. Let
illustration not visible in this excerpt
(3.5)
illustration not visible in this excerpt [ illustration not visible in this excerpt ]
where[ illustration not visible in this excerpt ]is generated by an ARMA process (Pantula, 1986).^{[29]} This model assumes that the conditional variance of the random disturbance depends linearly on the past behaviour of the squared errors, causing a positive autocorrelation in the volatility process or conditional variance, σt 2 , with a rate if decay governed by α + β, denoting that the closer α + β is to 1, the slower the decay of the autocorrelation of[ illustration not visible in this excerpt ]and shocks to the conditional variance are highly persistent, and the conditional variance is probably integrated (Engle and Bollerslev, 1986). The constraint[ illustration not visible in this excerpt ]permits the existence of a stationary solution, while the upper limit[ illustration not visible in this excerpt ]corresponds to the case of integrated process. In the GARCH model the variable of the mean specification,[ illustration not visible in this excerpt ]is a random component. It represents the innovation in mean of[ illustration not visible in this excerpt ]. This random disturbance or shock component can be interpreted as the single-period-ahead forecast error.
The volatility term structures of GARCH (1,1) allows to fix a long-term at a level that reflects any real scenario and use the Equation (3.5) to calculate the GARCH effect and persistence parameters using historical data:
illustration not visible in this excerpt (3.6)
Hence, the expectations consists that the time series change randomly about its mean. The error of the regression is normally distributed and heteroscedastic. This means that the variance of the current period’s error depends directly on the occurrence of a progress in the preceding period[ illustration not visible in this excerpt ]. The variance of[ illustration not visible in this excerpt ]is presented by the symbol illustration not visible in this excerpt . The variance equation describes how the error variance behaves.
illustration not visible in this excerpt (3.7)
Term structure estimations that are constructed with GARCH models mean-revert to the long term level of volatility at a speed that is determined by the estimated GARCH parameters. This property of GARCH models can lead to spurious empirical results when financial turmoil affects the market structure, generating high levels of volatility persistence.
Due to the fact that the inclusion of new information induces asymmetric GARCH effects, it is recommended to apply later the Glosten-Jagannathan-Runkle GARCH (GJR-GARCH) model by Glosten, Jagannathan and Runkle (1993) that models asymmetry in the GARCH process.
The Glosten-Jagannathan-Runkle GARCH (GJR-GARCH) model constructed by Glosten, Jagannathan and Runkle (1993) allows the conditional variance to react differently to the past negative and positive innovations (Glosten et al., 1993). In an application of different versions GARCH models used to estimate the daily Japanese stock return data, Engle and Ng propose using a news impact curve (Engle and Ng, 1995). The GARCH processes capture the characteristics of volatility clustering in the financial returns, but such models are unable to capture the leverage effect,^{[30]} that reflects the trend of negative correlation between current returns and future return volatility (Black, 1976).
The GJR-GARCH is an asymmetric or leverage model, in which “good” and “bad” news have a defined predictability for future volatility.^{[31]} By considering the presence of informational asymmetries in the market, commodity price changes may reveal information only to some agents. For example, low prices for a financial asset may be interpreted by an uninformed agent while better informed agents are not buying the asset or may be selling the asset. The evidence of informational asymmetries can have a serious impact on the performance of the real economy, causing an economic disequilibrium and thus volatility, in which the expectations of the agents concerning the commodity price returns do not match with each other. An extended explanation of the impact of information on the volatility based on the general equilibrium theory is not in the scope of this study. I will rather discuss the idea that past return shocks on the return volatility induce current parameters of the GARCH process, affecting the structure of the stochastic volatility models. Next section entails an examination of the volatility persistence based on the regime switching.
The existence of asymmetric information in the futures market leads to corresponding price shocks that correlate with the increase of volatility. News from other markets has a positive or negative effect on the oil price returns. In this case, the information from other market represents a relevant feature to be considered. The news is considered as given, and the transmission channels between the financial markets can be analysed with the implementation GARCH methodology.
The GJR-GARCH model focuses on the asymmetric effect of news on volatility. Specifically this asymmetric model provides a partially nonparametric model to reveal the empirical relationship between news and volatility. I adopt this asymmetric model to provide diagnostic tests referring to the relations between news and volatility. The measure of the impact of information on the volatility of oil price returns is required for a better estimation of conditional volatilities. Engle (1995) compares the properties of different volatility models to differentiate between the impact of “good news” and “bad news” on the conditional variance by analyzing the news impact curve of the ARCH classes.^{[32]} This asymmetric model is an extension of the GARCH process. The suggestion of the GJR (1,1) model may be expressed with the conditional variance:
illustration not visible in this excerpt (3.8)
where I (.) represents the indicator function, thus illustration not visible in this excerpt
illustration not visible in this excerpt where vt is i.i.d. The GJR model succeeds in measuring the relative influence of the so called “bad news”, i.e.[ illustration not visible in this excerpt ], and “good news”, i.e.[ illustration not visible in this excerpt ], on the volatility. One particular characteristic of this model is that it can be expressed in a regime-switching method:
illustration not visible in this excerpt (3.9)
where if[ illustration not visible in this excerpt ]the news impact curve is more perpendicular for negative shocks. The news impact curve is defined as a function of the current volatility that depends on most recent previous returns,[ illustration not visible in this excerpt ], and is considered as the dominant factor in determining price.^{[33]}. This model is also defined as a Sign-GARCH model.^{[34]} The GJR model is similar to the Threshold GARCH, or TGARCH model proposed by Zakoian (1994) and the Asymmetric GARCH, or AGARCH model of Engle (1990). According to Bollerslev (2008) the estimation of the GJR model with equity index returns, the parameter, γ , is typically obtained as positive, so that volatility increases proportionally more after a negative than positive shocks.^{[35]} This phenomenon is explained by the leverage effect.
ARMA model with GJR-GARCH innovations yields the ARMA(p,q)-GJR-GARCH(h,K) model for[ illustration not visible in this excerpt ]:
illustration not visible in this excerpt (3.10)
where[ illustration not visible in this excerpt ]in the mean formula of Equation (3.10) represents the i.i.d. standard normal or standardized Student- t random variable.
The standardized residuals from estimated models[ illustration not visible in this excerpt ]in several financial time series exhibit non-normal Gaussian distribution, that is, excess kurtosis which indicates departure from conditional normality. Therefore the fat-tailed distribution of the innovation causing an ARCH effect can be modelled using the Student’s- t or the Generalized Error Distribution (GED).^{[36]}
By taking the square root of the conditional variance and expressing it as an annualized percentage, the model yields a time-varying volatility estimate. The application of a simple estimation is needed to generate volatility model forecasts across any time period.
Financial time series, such as oil price, occasionally present sudden structural breaks, related to abnormal events such as financial distresses (Jeanne, Oliver et al., 2000), or unexpected changes in government economic policy. In these periods the economic variables associated with fundamental events behave quite differently during financial turmoil. In the following estimation section, it will be observed that an abrupt change of volatility state exhibited in the financial markets as consequence of the subprime crisis in 2008 had an impact on the development of the oil price curve, transferring the long-run volatility tendency from one market to another. Thus, the formation mechanism of the oil price suffered significant distortions during a period of high market uncertainty.
Furthermore, the principal objective in this section is to introduce the regime switching probabilities which captures probabilities of sudden returns changes and how persistent these changes in the behaviour of the oil price returns[ illustration not visible in this excerpt ]in order to empirically detect the volatility state of the oil market during a certain time period.
In the financial econometrics literature the Regime-switching models permits the system’s dynamics to switch between various states (in this case only two: high and low volatility state). With the application of this model I want to allow for periodic shifts in the parameters that indicate the underlying dynamics of volatility, in order to analyse the structural breaks in the given financial data. A common empirical finding is concerned with the fact that GARCH models tend to assign a high degree of persistence to the conditional volatility. This means that shocks to the conditional variance in the distant past still have an effect on the current time series.
The Markov-switching model estimation is superior to the basic GARCH model in at least two mean aspects: According to Lamoreux and Lastrapes (1990), the Markov-switching model explicitly accounts for the possibility of regimes shifts, whereas GARCH models do not. Secondly, the Markov-switching model can decompose the shock into two components: permanent shock and transitory shock.^{[37]} Regime changes influence the source of persistence in the conditional variance oil price returns.
By considering the introduction of regime shifts in both mean and variance structures, I will carry out a direct test of the link between the volatility of oil prices and its uncertainty over different time periods.
In the framework of MSGARCH models, a hidden Markov process { st } with state space {1, …, K } permits for discrete changes in the GARCH parameters. Following the seminal work of Hamilton and Susmel (1994), different parameterizations have been proposed to account for changes in the scedastic function’s parameters. However, these specifications lead to computational difficulties. The evaluation of the likelihood function for a sample of length T requires the integration over all KT possible paths, rendering the Maximum Likelihood (ML) estimation infeasible. Approximations are thus required to shorten the dependence on the state variable history but these lead to difficulties in the interpretation of the variance dynamics in each regime.
In order to avoid these problems, Haas et al. (2004) hypothesize K separate GARCH (1,1) processes for the conditional variance of the MSGARCH model. The conditional variances at time t can be written in vector form as follows:
illustration not visible in this excerpt (3.11)
where illustration not visible in this excerpt denotes the Hadamard product, (i.e. element-by-element multiplication). One conceptual advantage of the application of Markov-switching model in GARCH processes is that it captures the differences in the variance dynamics in low- and high-volatility periods. A relatively large value of[ illustration not visible in this excerpt ]and relatively low values of[ illustration not visible in this excerpt ]in high-volatility regimes may suggest a tendency to over-react to news, in comparison to other regular periods, while there is less memory in the sub-processes (Haas et al., 2004).
The central assumption of the Markov-switching model is that the market conditions can be differentiated into K regimes. Therefore it is assumed that the random variable[ illustration not visible in this excerpt ]switches according to the transition state or unobserved variable illustration not visible in this excerpt , which is a function of past values illustration not visible in this excerpt . In the analysis of oil and financial markets volatility it will be assumed that K = 2. For instance, if[ illustration not visible in this excerpt ], the process is in regime 1 at time t, and if[ illustration not visible in this excerpt ], the process is in regime 2 at time t. The advantage of this model to estimate the dynamics of oil prices is that they are significantly different between periods of economic „contraction“ and „expansion“. By detecting the volatility regime of the series, it is possible to estimate the degrees of volatility persistence in a given data series. This section focuses on the oil price returns by implementing models with volatility switching.
The observed conditional variance of financial time series (e.g., oil price and Standard and Poor’s Compounded Index returns), undergo changing periods of tranquillity and turbulence conditional to the recent information set,[ illustration not visible in this excerpt ]. Hence, financial asset prices show volatility clustering in which price changes[ illustration not visible in this excerpt ]at[ illustration not visible in this excerpt ]tends to be followed by a price change[ illustration not visible in this excerpt ]at t +1. This volatility clustering evinces that conditional variance of financial asset returns is non-stationary. In the previous section I have presented the ARCH and GARCH processes which are the most used approach for modelling conditional volatility. However the ARCH and GARCH models empirically tend to impute an excessive persistence to the conditional volatility in financial time series. This empirical finding implies that shocks to the conditional variance that occurred in the remote past keep on having non-trivial effects on the current variance. However, Lamoreux and Lastrapes (1990) indicate that high degrees of persistence to the conditional volatility may be spurious if there are structural breaks or regime shifts in the volatility process. Empirically, energy commodity prices like crude oil sufferered from large sudden price fluctuations caused partially by the 2008 global financial distress, which lead to exaggerated persistence in the oil price volatility.^{[38]} This persistent volatility occurs stochastically in a given business cycle period and is governed by a Markov process. The Markov-switching model of Hamilton (1989) and Sclove (1983) is determined by an unobservable state variable, which is modelled as a Markov chain.^{[39]} The following example shows the mean structure of the model:
illustration not visible in this excerpt (3.12)
where illustration not visible in this excerpt re presents an unobservable two-state Markov chain with the probability matrix P. It can be denoted that the different timing convention at time t is conditional to the hidden variable[ illustration not visible in this excerpt ]. In both states or regimes,[ illustration not visible in this excerpt ]implies the existence of an AR(1) process, with the difference, that the parameters (including the variance of the error term) varies across regimes, and the variation in regime is stochastic and probably serially correlated.^{[40]} The mean assumption of this model is that the regime is determined by st, the model rarely identifies in which regime the times series is, but after the identification of the regime the unobservable variable illustration not visible in this excerpt can be estimated. The assumption that the probability that[ illustration not visible in this excerpt ]equals j (=1 or 2 in this case) depends on the most recent or first order value[ illustration not visible in this excerpt ]
illustration not visible in this excerpt (3.13)
The basic form of Hamilton’s models (Hamilton, 1989) denotes an unobserved state variable illustration not visible in this excerpt , that is supposed to be estimated according to the first order Markov process. The transition probability[ illustration not visible in this excerpt ]is the probability that state i will be followed by state j. Thus, this regime switching transitional can be expressed as
illustration not visible in this excerpt (3.14)
Because probabilities sum to unity, then it results that
illustration not visible in this excerpt
Where illustration not visible in this excerpt and illustration not visible in this excerpt defines the probability of being in regime one, assuming that the system was in regime one during the past period, and probability of being in regime two, assuming that the system was in regime two.
Hamilton and Susmel (1994) and more recent studies such as Edwards and Susmel (2003) suggest that an almost integrated behaviour of volatility could be generated by the existence of structural changes (Soriano et al., 2006). Due to this feature of the time-varying volatility Hamilton and Susmel (op.cit.) introduced GARCH models with varying regime. Thus, the ARCH parameters change proportionally to a state or regime matrix of the variable in the previous period. One of the most important specifications of the SWARCH model is that the parameters of the underlying ARCH process can change with the time as a function of a previous information set.
The Markov-switching specifications of the SWARCH (K,q) model are able to detect endogenously different regimes of conditional variance. This model was proposed by Hamilton and Susmel (1994) and used to estimate interest rate volatility by Edwards and Susmel (2001). By applying the Markov-switching specification of the autoregressive conditional heteroscedasticity it permits discrete shifts and changes of persistence in the ARCH and/or GARCH parameters which react to small and large shocks. Furthermore, this specification allows estimation of a transitional probability pij, which represents the probability of a transition into state j when in state i, and magnitude of volatility at each state. A simple model to detect volatility clusters in financial time series is the ARCH model (Engle, 1982). The ARCH process is represented by the specifications of conditional mean and variance:
illustration not visible in this excerpt
illustration not visible in this excerpt (3.15)
illustration not visible in this excerpt
with[ illustration not visible in this excerpt ]An alternative parameterization of this model is the switching parameter which was presented by Hamilton and Susmel (1994) into parameterization:
illustration not visible in this excerpt (3.16)
where i =1,2 ,.., q and the transition or scale parameters that denote the change in regime, altering the squared errors and the conditional variance. Thus, g 1 is normalized to 1 because one of the g s is not unknown. The value of g captures the magnitude of conditional variance in state 2 relative to the state 1. Both parameterizations are equivalent if[ illustration not visible in this excerpt ]. Nevertheless, they generate different conditional variance processes if[ illustration not visible in this excerpt ]is time dependent. Thus, the switching ARCH model results from permitting time dependence of[ illustration not visible in this excerpt ]through a hidden K -state Markov chain st:[ illustration not visible in this excerpt ]
[...]
^{[1]} Excessive market volatility can negatively affect the real economic activity at all levels of the industry and, thus, can have an adverse effect on producers and consumers, Mabro, 2005, p. 25.
^{[2]} The increasing global integration of the major financial markets and the commoditization of crude oil have generated interest in examining the transmission of financial markets shocks across commodity markets, particularly crude oil.
^{[3]} Univariate SWARCH models were applied by the IMF researchers to indentify the high or low volatility periods during the financial turmoil after the Lehman Brothers failure. International Monetary Fund. Financial Stability Report April 2009, p. 136.
^{[4]} The International Monetary Fund has used Markov-switching models to determine the degrees of systemic risk in the financial markets. Source: International Monetary Fund. Financial Stability Report, April, 2009, p. 111-145.
^{[5]} Jalali-Naini, 2007, p. 45.
^{[6]} The available data for the futures market is based on two traded benchmarks: WTI and Brent crude. These benchmarks represent light and sweet crude, but they are not representative for the large volume of heavy and sour oil traded physically. However, it is convenient in the scope of this research to choose the WTI crude because the volume of traded contracts is larger than the physical representative oil crude (i.e. Dubai Mercantile Exchange Oman Crude Oil Futures Contract).
^{[7]} Ewinga et al., 2002, p.526.
^{[8]} Alexander, 2001, p.4.
^{[9]} Ibid.
^{[10]} «A speculator is a trader or investor who enters in the futures market in search of profit, by doing so, willingly acess increased risk.» Kolb, 1997, p.154.
^{[11]} World Economic Outlook, October 2008: Financial Stress, Downturns, and Recoveries von International Monetary Fund von International Monetary Fund, p.119.
^{[12]} Jalali-Naini, 2009, p. 41.
^{[13]} Mabro, 2005, p.12-13.
^{[14]} Fleming (1998) constructed a model that shows how cross-market hedging and sharing of common information could lead to transmission of volatility across markets over time.
^{[15]} As expressed above, financial contagion assumes that a significant increase of correlation coefficients of the cross-market linkages is the reaction of a shock in an individual market.
^{[16]} Implied volatility is defined as the volatility of the underlying price asset process that is implicit in the market price of an option. This volatility is computed with the call option pricing formula Black-Scholes for European optionswith the call option’s market value (Black and Scholes, 1973). Implied volatility is a forecast of the process volatility, because it refers to the market's assessment of future volatility.
^{[17]} A call option is defined as a financial contract between two parties: the buyer and the seller of this type of option. It is the option but not the obligation to buy shares of stock at a specified time in the future.
^{[18]} Hol, 2003, p.13.
^{[19]} Soofi and Liangyue, 2002, p. 121.
^{[20]} Alexander, 2001, p.63.
^{[21]} Pilipovic, 2007, p. 224.
^{[22]} Sadorsky, 2006, p. 471.
^{[23]} Heij et al., 2004, p. 621-622.
^{[24]} Ibid.
^{[25]} Alexander, 2001, p. 71.
^{[26]} Ibid.
^{[27]} Alexander, 2001, p. 111.
^{[28]} Adrangi et al. (2001) define the nonlinearities as the second and higher order dependence between the energy prices.
^{[29]} Bollerslev, 1986, p. 310.
^{[30]} The explanation of the leverage effect in equity markets is based on the fact that negative returns raise a firm’s financial leverage which increases its risk and therefore equity volatility (in terms of the debt-to-equity ratio). However
^{[31]} Black, 1976, p. 145.
^{[32]} Engle, 1995, p. 147.
^{[33]} Franke et al., 2008, p. 305.
^{[34]} Bollerslev, 2008, p. 16.
^{[35]} Ibid.
^{[36]} Nelson (1991) suggested to apply the generalized error distribution (GED) to capture the leptokurtic distribution commonly observed in the distribution of financial time series.
^{[37]} Bhar et al., 2004, p.107.
^{[38]} There are many economic and econometric studies which support the hypothesis that the excessive speculation in the futures market caused an excessive demand for futures papers causing an increase in the price of oil. For more information see Jalali and CFTF documents about the regulation of the futures market.
^{[39]} Campbell et al., 1997, p. 472.
^{[40]} Ibid., p. 473.
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