Crude can identify sinusoidal components along with their amplitudes

Crude oil is one of the most financially liquid products that are being traded globally; therefore, an extensive literature about crude oil markets is available, and these researches are conducted by financial, academic, and government institutions. Fundamentals of crude oil, such as supply and consumption, are key indicators for local and global economies; therefore, the relation between price drivers of crude oil and macro variables was tackled in many researches. In this section, literature on crude oil markets is presented to elaborate on what has been done until today.Secondly, wavelet analysis is a relatively new concept that is used at various fields of science and engineering. The literature review of wavelet analysis illustrates multi-disciplinary approach of academics, and it is evident that wavelet analysis has significant contribution to literature due to its unique properties.2.1 WAVELET ANALYSIS IN FINANCENowadays, finance world is interested in components of the price series rather than the price series themselves. Wavelet is the most recent and robust tool, which is able to decompose the price series into scales. In other words, wavelet analysis is an extended version of spectral analysis that examines frequency domain. Therefore, physicists, software engineers, and mathematicians have been involved in finance sector for several decades. Although wavelet transformation is recently developed, mathematical functions have been investigated in terms of sinusoidal components since Joseph Fourier. Before Joseph Fourier, 18th century mathematician, Leonhard Euler, opened up a new era in mathematics by discovering the concept of Euler’s identity. The concept is used to transform sinusoidal functions into exponential form, and the equation was later called Euler’s formula. The technical details of the formula are explained at the quantitative methods section of the thesis.Joseph Fourier transformed components of a time domain function into components at frequency domain in 19th century. This is known as Fourier transform, and the transformation process can identify sinusoidal components along with their amplitudes from the signal. However, this methodology assumes that all components of the signal6do not change over time. In other words, jumps, seasonalities, and trends in the signal cannot be captured by the transformation. Time localization of sinusoidal components is required to accurately analyze non-stationary signals.Dennis Gabor (1946) inserted a rolling window into the model to localize the sinusoidal functions in time, which is called short term Fourier transformation (STFT). His solution is to apply the transformation within a rolling window. However, according to Heisenberg’s uncertainty principle; the window cannot achieve to localize time and frequency simultaneously.In the beginning of 20th century, Alfred Haar found first simple wavelet with finite energy in his study on orthogonal functions. Then, Jean Morlet, who is an engineer working for an oil company, found a new way to localize the components in time and frequency domains, and named it as “wavelets of constant shape” (Mackenzie, 2001). He simply kept the shape of the wavelet same while dilating, compressing, and shifting.Yves Meyer introduced orthogonal wavelets, which can extract information contained in the scale-based components of a signal, and his student, Stephane Mallat, combined the wavelet theory with multi-resolution analysis in 1986 (Mackenzie, 2001). By the end of 20th century, Ingrid Daubechies used Mallat’s studies to develop different set of basis functions that are used in this thesis.Currently, wavelets have a variety of applications which includes the analysis of the relation between several time-series, forecasting time-series, and the analysis of time-variant volatilities. These applications are widely included in studies of finance and economics, because most of the financial and economic series are non-stationary; so robust time and frequency localization properties are required while using spectral analysis tools to decompose the financial time series into components.Conventional statistical models can be applied to wavelet decomposed series, which are components of the original signal and which are obtained without losing any information. For instance, Granger causality test can be applied to decomposed series after wavelet transformation. Ramsey and Lampart (1997) analyzed the relationship between wavelet decomposed series of money supply and income by using Granger causality test. They used Symlet as basis function because symmetry of the function7was a major determinant for their research results. Moreover, they applied Granger causality test to reconstructed series of each scale, as well as to log-differences of original series. They found different causality relation for each scale and for each period. Furthermore, in spite of the fact that they found causality relation among scales, they did not find any causality relation at log-differences of original series.Wavelet analysis is a very powerful tool to analyze the behavior of non-stationary time series. Financial time series have complex structure, and they include shifts, jumps, cycles and trends. From spectral point of view, they have components that appear and disappear within the life time of the signal and require multi-resolution analysis. Ramsey, Usikov and Zaslavsky (1995) were early researchers on this topic. Their study on stock markets shows that financial data is predictable. They found evidence about quasi-periodicity in the occurrence of large amplitude shock.Uddin, Tiwari, Arouri and Teulon (2013) analyzed the co-movement of oil prices and Japanese Yen by using wavelet analysis. They used continuous wavelet transform, cross wavelet, and wavelet coherency techniques. Moreover, they improved bias in the wavelet power spectrum. They found a strong relation between real oil prices and real exchange rates for the short-term and mid-term time horizon. They stated that this relation is meaningful for noise traders and high frequency traders rather than fundamentalist traders.In literature, wavelet covariance between financial assets is also used to set up better diversified portfolios that have minimum risk. Conlon and Cotter (2012) analyzed hedging effectiveness of OLS hedging model applied to wavelet decomposed time-series. They aimed to capture minimum-variance hedge ratio and aimed to solve dynamic hedging time horizon problem through a moving window. They used variance, as well as value-at-risk, to assign different weights to positive and negative returns, and to measure hedging effectiveness. They preferred MODWT for multi-resolution analysis and selected crude oil, S&P 500 index and GBP/USD futures for the portfolio. They calculated variance-minimizing hedge ratio from wavelet coefficients of each scale. They found that the model performs better for reducing variance than reducing value-at-risk. They concluded that effectiveness of the model increases for longer time-horizons.8Conlon and Cotter, along with Gencay (2012), did a second study about commodity hedge efficiency of scale based hedge ratios. The contribution of this second study is the inclusion of expected utility function in order to obtain the optimal hedge ratio. The utility function takes investor preference about risk aversion as input; therefore the model gives same result with the conventional variance-minimizing model if risk aversion parameter is set to ?. They preferred Symlet as wavelet scaling function to be used with MODWT. They found that best performing model in terms of risk reduction is the risk averse model; on the other hand, best utility maximizing model has lowest risk aversion parameter.Furthermore, Maharaj, Moosa, Dark and Silvapulle (2008) analyzed hedge efficiencies of crude oil, soybeans, and S&P 500 index through MODWT. The paper was organized around two main purposes, which are analyzing how preference of asymmetric hedging methodologies improves effectiveness, and analyzing the relation between hedge horizon and data frequency. However, unlike most of the studies in the literature, they concluded that wavelet analysis does not contribute to hedging effectiveness of the models.Michis (2014) conducted a study to evaluate individual risk contribution of gold to a portfolio composed of several assets. He used MODWT to calculate variance-covariance matrix of each scale. Partial derivatives of portfolio variance formula are taken to calculate individual risk contributions of each asset to the portfolio. He found that volatility contribution of gold is significant in lower scales; however the risk decreases sharply in higher scales. Furthermore, three month treasury-bill has the lowest risk contribution in short term; and stock risk in long term is significantly lower than the risk in short term.One common use of wavelets is removing noise from the original signal to obtain more smooth series. As an alternative, this can be achieved through frequency domain filters by removing the components that has specific frequencies. However, wavelets transformation is able to localize noise in time. In literature, wavelet denoising is mostly used to improve accuracy of forecasting models, and is also used to analyze the time varying behavior of the time series, such as volatility.9Dremin and Leonidov (2006) applied universal threshold wavelet filtering to stock prices in order to test how volatility autocorrelation function changes. They found that denoised series are less Gaussian, and volatility autocorrelation function is almost similar with the function of original series.Zhang, Coggins, Jabri, Dersch and Flower (2001) designed a hybrid model by using wavelet analysis and neural networks, named neuro-wavelet hybrid system, to forecast prices of futures contracts that have four different maturity dates. Basically, they transformed the time series into wavelet coefficients, and fed a number of neurons with the coefficients. The neurons forecast wavelet coefficients for each scale. To overcome translation invariance problem of DWT, they used autocorrelation shell representation to convert predicted coefficients into original series. Compared with conventional multi-layer perceptrons, they doubled the profit, and increased the Sharpe ratio.Mingming and Jinliang (2012) developed a hybrid model for forecasting. Their model uses recurrent neural network and wavelet analysis to forecast oil prices. The model takes historic oil prices and gold prices as inputs, and generates short-term prediction. They used DWT along with Daubechies wavelet filter to identify non-linearity; as a result, they obtained forecast results with low error rates.Finally, Ramsey (1999) studied on the literature of wavelet usage in finance and economics. Moreover, he pointed out the differences between denoising and noise smoothing. He suggested that linkages between the scales should be considered for further studies.2.2 CRUDE OIL AS A FINANCIAL PRODUCTAccording to CME leading products report of Q4, 2015, crude oil futures contracts are the most traded commodity products. Hedgers, arbitrageurs, and speculative traders are interested in taking positions at crude oil markets due to various purposes. For instance, speculative traders take the price risk from hedgers to make profit from their forecasts. Therefore, statistical properties of the underlying product and futures contract are important for the hedgers and the traders to estimate exposed risks and potential opportunities. The existing finance literature is sufficient to understand price dynamics of crude oil markets and to understand the interaction of crude oil prices with other10markets. In this section, literature about crude oil as a financial product is investigated which may enable the reader to recognize the contribution of this thesis to existing literature.Pindyck (2004) analyzed the time varying volatility of both natural gas and oil prices after 1990. According to his study, there was no volatility increase after Enron collapse. Moreover, it is important to point out that he found mean-reversion at volatility of natural gas and oil prices. Based on his summary, he stated that the volatility shocks decreases in 5-10 weeks, and he came up with a result of short-lived volatility shocks.Tabak and Cajueiro (2007) analyzed the efficiency of crude oil products, which are WTI and Brent. They estimated the time varying Hurst ratio, which decreases by the time; and they concluded that crude oil prices have long memory.Caporale, Ciferri and Girardi (2010) analyzed the price discovery of spot and futures prices of crude oil. They used cointegration test, and they found that long-term relation between spot prices and nearest term futures contracts is strong; however, as maturity of futures contracts increases, the relation weakens. Moreover, they estimated contributions of the products to price discovery process through feedback matrix. The results show that largest portion of price discovery is handled by futures markets if nearest-term futures contracts are considered.There is an extensive literature about the cointegration, causality and lead-lag relation between spot and futures prices. Bekiros and Diks (2008) applied Granger causality test to log returns of WTI spot and futures contracts that have maturity of one, two, three and four months. They found uni-directional linear relationship between the spot and the futures contract that has maturity of four months. The remaining contracts have bi-directional relationship with the spot prices; and they found that if nonlinear effects are controlled, the lead-lag relationship is not permanent over time.The information obtained from the relation between spot and futures markets of crude oil is used by hedgers to eliminate the price risk that they are exposed to. Governments that are either exporter or importer of oil products are obligated to provide the required cash-flow to meet their projected budget. Therefore, a fluctuating crude oil price is a risky item in their cash stream that requires some mitigating actions. Daniel (2001)11recommended that these countries should hedge their oil price exposure via financial tools despite its political difficulties.Producers and consumers can eliminate the price uncertainty through available hedging channels; such as futures and forward markets. They can fix the fluctuating price of the commodity for a predetermined time in the future and predetermined volume to deliver or to receive. However, the literature shows that the relation between spot and futures prices is not linear and constant. Moreover, one unit of price change in spot prices may yield a different unit of price change in futures prices, or vice versa. Therefore, natural long or short spot positions cannot be perfectly hedged by just taking opposite positions at futures markets. Optimal hedge ratio models aim to find out the optimal hedge volume that protects the natural long/short positions of the hedger. Furthermore, dynamic hedging models in literature solve the time-varying relation problem.The price risk of the commodity is quantified by its variance; therefore future variance of the product is the focus of the hedger to eliminate the price risk. One common method is to consider the variance as constant, and using historic variance as a risk measure of the commodity prices. However, better hedge efficiencies are achieved by modeling time varying distributions of the prices. Moreover, future joint distribution of spot and futures prices is important to achieve a better hedge efficiency. Conditional correlation modeling techniques are used to forecast future joint distributions of spot and futures prices.Chang, McAleer, and Tansuchat (2009) used conditional correlation models of CCC and DCC, and various versions of GARCH to forecast variances and correlations of spot and futures prices for Brent, WTI, and Dubai crude oils. Lanza, Manera, and McAleer (2006) also used DCC for WTI spot, futures, and forward prices. They stated that negative shocks affect the volatility greater than the positive shocks. They found that conditional correlation between WTI forwards and futures prices varies by the time.McAleer (2010) compared hedge performances of strategies by using different models, which are CCC, VARMA-GARCH, DCC, BEKK, and diagonal BEKK, for Brent and WTI crude oil markets. He found that the hedge ratios derived from diagonal BEKK are the best risk eliminating ratios among the ratios estimated from other models.