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Hedging with Commodity Futures

Name: Hedging with Commodity Futures
Price: 47.95 EUR
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Author: Su Dai
ISBN: 978-3-656-54017-5

Master's Thesis, 2013

77 Pages, Grade: 1,7

Su Dai (Author)

Excerpt

List of Tables

List of Figures

List of Abbreviations

1 Introduction

2 Hedging Strategies
2.1 Risk Measures
2.1.1 Standard Deviation and Its Variance
2.1.2 VaR and CVaR
2.2 Optimization of Hedging Objective Functions
2.2.1 Minimum Variance Hedge
2.2.2 Minimum (C)VaR Hedge

3 Modeling Conditional Return Distribution
3.1 Relationship between Futures and Spot Prices
3.2 GARCH Models
3.2.1 GARCH (1,1)
3.2.2 Elliptical Distribution
3.2.3 Multivariate GARCH
3.3 Regime Switching Models
3.3.1 Markov Chains
3.3.2 Mixture of Distributions
3.3.3 MRS-Model
3.3.4 MRS-GARCH Model

4 Implementation
4.1 Description of the Data and Their Properties
4.1.1 Testing for Normality
4.1.2 Testing for ARCH Effects
4.1.3 Testing for Stationarity
4.1.4 Testing for Cointegration
4.2 Parameter Estimation
4.3 Estimation Results
4.4 Hedge Ratios
4.5 Hedging Effectiveness of the Hedging Strategy

5 Conclusion and Future Outlook

Appendix A

Appendix B Matlab Codes

Bibliography

Ehrenwörtliche Erklärung

List of Tables

Table 1 Descriptive Statistics for Daily Spot and Futures Prices

Table 2 Jarque-Bera Test Results

Table 3 Ljung-Box Test Results

Table 4 ADF Test

Table 5 Testing for Cointegration

Table 6 Estimates of Single-Regime BEKK GARCH Models under Assumption of Normal and Student´s-t Distribution

Table 7 Estimates of Single-Regime CCC GARCH Models under Assumption of Normal and Student´s-t Distribution

Table 8 Estimates of Markov Regime Switching Model

Table 9 Estimates of MRS-BEKK Model

Table 10 Comparison of OLS Hedge Ratios

Table 11 Comparison of Hedge Ratios

Table 12 Hedging Effectiveness of Dynamic Hedging Models against the Constant OLS Model

List of Figures

Figure 1 Smooth Regime Probabilities

Figure 2 Smooth Regime Probabilities

Figure 3 OLS and Rolling Window OLS Hedge Ratios

Figure 4 MV-Hedge Ratios

Figure 5 Min-VaR Hedge Ratios

Figure 6 Min-CVaR Hedge Ratios

List of Abbreviations

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1 Introduction

The commodity futures contract is an agreement to deliver a specific amount of commodity at a future time^[1]. There are usually choices of deliverable grades, delivery locations and delivery dates. Hedging belongs to one of the fundamental functions of futures market. Futures can be used to help producers and buyers protect themselves from price risk arising from many factors. For instance, in crude oil commodities, price risk occurs due to disrupted oil supply as a consequence of political issues, increasing of demand in emerging markets, turnaround in energy policy from the fossil fuel to the solar and efficient energy, etc. By hedging with futures, producers and users can set the prices they will receive or pay within a fixed range.^[2] A hedger takes a short position if he/she sells futures contracts while owning the underlying commodity to be delivered; a long position if he/she purchases futures contracts.^[3]

The commonly known basis is defined as the difference between the futures and spot prices, which is mostly time-varying and mean-reverting. Due to such basis risk, a naïve hedging (equal and opposite) is unlikely to be effective. With the popularity of commodity futures, how to determine and implement the optimal hedging strategy has become an important issue in the field of risk management.^[4]

Hedging strategies have been intensively studied since the 1960s. One of the most popular approaches to hedging is to quantify risk as variance, known as minimum-variance (MV) hedging.^[5] This hedging strategy is based on Markowitz portfolio theory, resting on the result that “a weighted portfolio of two assets will have a variance lower than the weighted average variance of the two individual assets, as long as the two assets are not perfectly and positively correlated.”^[6]

MV strategy is quite well accepted, however, it ignores the expected return of the hedged portfolio and the risk preference of investors. Other hedging models with different objective functions have been studied intensively in hedging literature.^[7] Due to the conceptual simplicity, the value at risk (VaR) and conditional value at risk (C)VaR have been adopted as the hedging risk objective function.

Under the assumption that the joint distribution of the spot and futures prices is time invariant, the MV hedge ratio (MVHR) can be estimated using simple techniques such as Ordinary Least Squares (OLS). However, using OLS model for estimating optimal hedge ratio (OHR) has been challenged due to several reasons. Over the time many different techniques have also been used in estimating the OHR, the conditional variance and covariance of the spot and futures returns are required under the time-varying framework.

The Generalized Autoregressive Conditional Heteroscedasticity (GARCH) model and its enormous extensions and derivations have proved to be very useful in estimating jointly the conditional variances and covariances that is required in estimating the optimal hedge ratios, and they have demonstrated better performance when used to generate forecasts of the variance-covariance matrix over short time horizons.^[8] Within the literature that looks at the modeling of spot and futures returns in commodity markets, the use of multivariate GARCH (MGARCH) models for modeling conditional variances and covariances of spot and futures prices has become very common in recent years. This thesis follows recent literature by modeling the conditional (co)variance with MGARCH models.

As a comparison study to GARCH models, this thesis also uses Markov Regime Switching (MRS) models allowing the MVHR to be both time varying and state-dependent. MRS models introduced by Goldfeld and Quandt (1973) are a very recent development in the futures hedging literature. The rationale behind this model is that changes in market conditions may affect the relationship of spot and futures prices. The assumption of constant parameters throughout the estimation period is relaxed, this is done by allowing the volatility to switch stochastically between different processes under different market conditions, which may lead to some improvements of the ‘fit’ of the data.^[9]

The purpose of this thesis is therefore to estimate the MVHR and Min-(C)VaR hedge ratios using several econometric models and compare the empirical performance of these models. The rest of the paper is organized as follows:

Chapter 2 begins with a brief introduction of risk measures, then the hedging objective functions will be presented. GARCH and MRS models modeling conditional return distribution of both the spot and futures will be studied in Chapter 3. Chapter 4 proceeds to an empirical study and applies the selected models in a portfolio setting. The estimated results will be reported. Finally, this thesis uses pre-specified performance measures to compare the hedging performance of the selected models. Last chapter offers a conclusion and suggestions for future research.

2 Hedging Strategies

To test the hedging strategy, this thesis begins with the set-up, consider a crude oil extraction company wants to hedge against falling crude oil price by taking up a position in the crude oil futures market. This hedger employs a short hedge to lock in a future selling price for an ongoing production of crude oil that is ready for sale in the future. In order to simplify, scenarios for portfolios including more than one type of commodity position as well as cross-hedging options are ignored.

Let Abbildung in dieser Leseprobe nicht enthalten and Abbildung in dieser Leseprobe nicht enthalten denote the logarithm of spot and futures prices, respectively. The relationship between both prices at time Abbildung in dieser Leseprobe nicht enthalten is usually written as

illustration not visible in this excerpt

where Abbildung in dieser Leseprobe nicht enthalten is the risk free interest rate and Abbildung in dieser Leseprobe nicht enthalten is time to maturity of the contract. This relationship is derived from a non-arbitrage condition and is subject to certain assumptions such as no transaction costs, tax rate and possibility of borrowing and lending at the same risk free interest rate.

If a hedge has to be closed prior to maturity of the contract, the hedger is exposed to basis risk, which is the difference between the two prices.

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One-period spot and futures returns are:

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The one-period return on the hedged portfolio, Abbildung in dieser Leseprobe nicht enthalten, is given by:

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where Abbildung in dieser Leseprobe nicht enthalten is the portfolio weight in the futures positions and is the so-called hedge ratio. In naïve hedging, the hedge ratio equals to 1.

The hedge ratio is based on an available information set Abbildung in dieser Leseprobe nicht enthalten. If Abbildung in dieser Leseprobe nicht enthalten contains relevant information^[10], the hedging strategies become time-varying conditional hedging strategies. In contrast, if Abbildung in dieser Leseprobe nicht enthaltendoes not contain past observation, Abbildung in dieser Leseprobe nicht enthalten is independent of Abbildung in dieser Leseprobe nicht enthalten, the hedging strategies are time-invariant unconditional.^[11]

2.1 Risk Measures

When making hedging decisions, a hedger has to seek a balance between risk and returns. When minimizing risk, the choices of risk measures are very important. A risk measure is equivalent to establishing a correspondence Abbildung in dieser Leseprobe nicht enthalten between the space X of random variables, e.g. the returns of a given set of investments and a nonnegative real number, i.e. Abbildung in dieser Leseprobe nicht enthalten. Generally, any coherent risk measure must have the following properties:^[12]

(P.1) Positive homogeneity: Abbildung in dieser Leseprobe nicht enthalten for all random variables Abbildung in dieser Leseprobe nicht enthalten and all positive real numbers Abbildung in dieser Leseprobe nicht enthalten.

(P.2) Subadditivity: Abbildung in dieser Leseprobe nicht enthaltenfor all random variables Abbildung in dieser Leseprobe nicht enthalten and Abbildung in dieser Leseprobe nicht enthalten;

(P.3) Monotonicity: Abbildung in dieser Leseprobe nicht enthalten implies Abbildung in dieser Leseprobe nicht enthalten for all random variables Abbildung in dieser Leseprobe nicht enthalten and Abbildung in dieser Leseprobe nicht enthalten

(P.4) Transitional invariance: Abbildung in dieser Leseprobe nicht enthaltenfor all random variables Abbildung in dieser Leseprobe nicht enthalten and real numbers Abbildung in dieser Leseprobe nicht enthalten, and all riskless rates Abbildung in dieser Leseprobe nicht enthalten.

Some studies replace the first two conditions of coherent with the condition that Abbildung in dieser Leseprobe nicht enthalten be convex, i.e. that

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The axiom of convexity is motivated by the idea that diversification should not increase the risk.^[13]

Before starting with the description of risk measures, some fundamentals about multivariate statistics should be presented.^[14]

Consider a general Abbildung in dieser Leseprobe nicht enthalten-dimensional random vector of risk factor changes (for instance price changes or returns) Abbildung in dieser Leseprobe nicht enthalten The distribution of Abbildung in dieser Leseprobe nicht enthalten is completely described by the joint distribution function (DF).

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The marginal DF of Abbildung in dieser Leseprobe nicht enthalten, written Abbildung in dieser Leseprobe nicht enthalten or Abbildung in dieser Leseprobe nicht enthalten, is the distribution function of that risk factor considered individually. For all Abbildung in dieser Leseprobe nicht enthalten,

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Let Abbildung in dieser Leseprobe nicht enthalten be a partition of Abbildung in dieser Leseprobe nicht enthalten, where Abbildung in dieser Leseprobe nicht enthalten and Abbildung in dieser Leseprobe nicht enthalten, then the marginal distribution function of Abbildung in dieser Leseprobe nicht enthalten is

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The DF of a random vector Abbildung in dieser Leseprobe nicht enthalten is said to be absolutely continuous if:

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Where Abbildung in dieser Leseprobe nicht enthalten is a non-negative function, which is then known as the joint density of Abbildung in dieser Leseprobe nicht enthalten.^[15]

Let Abbildung in dieser Leseprobe nicht enthalten denote the joint density of the Abbildung in dieser Leseprobe nicht enthalten -dimensional marginal distributionAbbildung in dieser Leseprobe nicht enthalten, then the conditional distribution of Abbildung in dieser Leseprobe nicht enthalten given Abbildung in dieser Leseprobe nicht enthalten has density

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and the corresponding DF is

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The mean vector of Abbildung in dieser Leseprobe nicht enthalten, when it exists, is given by

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The covariance matrix Abbildung in dieser Leseprobe nicht enthalten, when it exists, is defined by

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where the expectation operator acts componentwise on matrices.

TheAbbildung in dieser Leseprobe nicht enthalten element of the matrix is

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This is ordinary pairwise covariance between Abbildung in dieser Leseprobe nicht enthalten and Abbildung in dieser Leseprobe nicht enthalten. The diagonal elements Abbildung in dieser Leseprobe nicht enthalten are the variances of the components of Abbildung in dieser Leseprobe nicht enthalten and Abbildung in dieser Leseprobe nicht enthalten are standard deviations of the components of Abbildung in dieser Leseprobe nicht enthalten.

The correlation matrix of Abbildung in dieser Leseprobe nicht enthalten, denoted by Abbildung in dieser Leseprobe nicht enthalten, can be defined by introducing a standardized vector Abbildung in dieser Leseprobe nicht enthalten such that Abbildung in dieser Leseprobe nicht enthalten for all Abbildung in dieser Leseprobe nicht enthalten and taking Abbildung in dieser Leseprobe nicht enthalten. The Abbildung in dieser Leseprobe nicht enthalten element of this matrix is

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2.1.1 Standard Deviation and Its Variance

Markowitz (1952) developed a mean-variance (MV) framework for portfolio selection in which he argued, that investors in choosing between portfolios with equal expected value, should choose the portfolio with the minimum variance. Since then, variance and standard deviation have become the most common risk measures in portfolio optimization. Variance measures the risk associated to the return of each investment, it is defined as the deviation from the mean of the return distribution.^[16]

The use of the variance and standard deviation as risk measures has been criticized. It makes no distinction between positive and negative deviations from the mean. This implies that a rational investor would dislike the potential gain to the same degree as the potential loss when optimizing his portfolio.^[17] However, positive and negative deviations of the returns from their mean do play a greatly asymmetric role in the investor’s perception.^[18] In fact, variance measures uncertainty, which, although related, is not the same as risk.^[19]

Furthermore, the standard deviation or variance underestimates the tail risk of the return distribution, especially when return distributions are not (approximately) symmetric around the mean. Therefore, it fails in capturing all of the characteristics of portfolio returns that investors consider to be important and is no longer an appropriate measure of risk.^[20] However, a considerable amount of applications of optimal hedging is still using the criterion of minimum variance.

2.1.2 VaR and CVaR

To overcome the problems stated above, financial practice and related theory show increasing interest towards downside risk measures. Knill et al. (2006) suggested that if an oil and gas company uses futures contracts to hedge risk, they hedge only the downside risk.^[21] Due to the conceptual simplicity, the value at risk (VaR) and conditional value at risk (CVaR) have been widely used as risk measures.

In portfolio setting, VaR is defined as “the maximum loss on a portfolio over a certain period that can be expected with a certain probability”.^[22] In terms of the formal definition, the VaR of Abbildung in dieser Leseprobe nicht enthalten at confidence level Abbildung in dieser Leseprobe nicht enthalten is the negative of the lower Abbildung in dieser Leseprobe nicht enthalten-quantile of the profit/loss distribution, where Abbildung in dieser Leseprobe nicht enthalten.^[23]

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where Abbildung in dieser Leseprobe nicht enthalten is the probability measure.Abbildung in dieser Leseprobe nicht enthalten is typically chosen between 1% and 5%.

If the loss distribution is continuous, then

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The relative VaR is defined as

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which is the difference between the expected value of Abbildung in dieser Leseprobe nicht enthalten and the lower quantile of Abbildung in dieser Leseprobe nicht enthalten.^[24]

Although VaR is a very popular measure of risk, it has undesirable mathematical characteristics. Under the assumption of elliptical distributions, VaR is subadditive.^[25] In this case, a VaR-minimizing portfolio coincides with the Markowitz variance-minimizing portfolio^[26]. If applied to non-elliptical return distributions, VaR is not an acceptable risk measure^[27]: first, it does not measure losses exceeding VaR and it may provide conflicting results at different confidence levels; second, non-subadditivity implies that portfolio diversification may lead to an increase of risk and prevents to add up the VaR of different risk sources; third, non-convexity makes it impossible to use VaR in optimization problems. Several studies showed that “VaR can be ‘ill-behaved’ as a function of portfolio positions and can exhibit multiple local extrema, which can be a major handicap in when determining an optimal mix of positions or even the VaR of a particular mix.”^[28]

An alternative coherent measure of risk is the CVaR. Similar concepts can be found in literature: Expected Shortfall, Tail Conditional Expectation or Tail-VaR. The definition varies across different scholars.

Abbildung in dieser Leseprobe nicht enthalten is given by

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when distribution function Abbildung in dieser Leseprobe nicht enthalten is continuous at Abbildung in dieser Leseprobe nicht enthalten.

CVaR can equivalently be expressed as a VaR average:

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Zhang and Rachev (2004), Rockafellar and Uryasev (2002) showed that the CVaR satisfies the axiom of coherent risk measures. The proof is omitted here.

As advocated by Artzner et al. (1999), CVaR is preferred in practice due to its better mathematical properties. In optimization modeling, CVaR can be expressed as a minimization formula suggested by Rockafellar and Uryasev (2000).

Despite the shortcomings, however, VaR is still widely used as risk measure, partly because of its adoption by the Basel Committee on Banking Regulation for the assessment of the risk of the proprietary trading books of banks and its use in setting risk capital requirements.^[29] Therefore, this thesis will consider both risk measures, the VaR and CVaR, for optimal hedging.

2.2 Optimization of Hedging Objective Functions

The optimal hedge ratio depends on the particular objective function to be optimized. This section will present three objective functions using the risk measures presented in Section 2.1.

2.2.1 Minimum Variance Hedge

Johnson (1960) and Ederington (1979) were the first to derive the minimum variance hedge ratio (MVHR) as the “average relationship between the changes in the cash price and the changes in the futures price”, which minimizes the net price change risk, where net price change risk is measured by the variance of the price changes of the hedged position.^[30]

The variance of the hedged portfolio is given by

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The first order condition (FOC) of portfolio variance with respect to Abbildung in dieser Leseprobe nicht enthalten:

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The portfolio variance is minimized when the FOC derived above is equal to zero, thus the MVHR is derived as follows:

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The hedge ratio is then the ratio of covariance of spot and futures returns to variance of futures returns.

Assume that the joint distribution of spot and futures returns is time-invariant, the minimum variance hedge vector Abbildung in dieser Leseprobe nicht enthalten can be then interpreted as the slope coefficient Abbildung in dieser Leseprobe nicht enthalten obtained by a linear regression of the change in the spot prices against the changes in the futures prices.^[31]. This is given by:

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Where Abbildung in dieser Leseprobe nicht enthaltenand Abbildung in dieser Leseprobe nicht enthaltencorrespond to spot and futures returns respectively for period Abbildung in dieser Leseprobe nicht enthalten and Abbildung in dieser Leseprobe nicht enthalten is the error term that follows a multivariate normal distribution with zero mean. Simple techniques such as Ordinary Least Squares (OLS) can be used for estimation.

Several empirical studies have employed the OLS model to estimate static hedge ratios and measures of hedging effectiveness.^[32] However, using OLS model for estimating optimal hedge ratio has been challenged: A major problem with the OLS hedge ratio is its dependence on the unconditional second moments.^[33] It is highly unlikely that the variance of the time series of return remains constant over time as new information arrives to the market the shape of the distribution changes. Moschini and Myers (2002) stated that a time-varying covariance matrix of cash and futures prices, per se, is not sufficient to establish that the optimal hedge ratio is time varying. However, their test results could clearly reject the null hypothesis of a time-constant hedge ratio for corn spot and futures price, for a time-varying hedge ratio.^[34] The MVHR might vary over time as the new information reaches the market.

2.2.2 Minimum (C)VaR Hedge

The Minimum-(C)VaR hedge ratio requires the hedger to decide his or her risk-averse level and hedging horizon, this is because different individuals choose different optimal hedge ratios, depending on the values of their own risk consideration and hedging horizon: more risk averse hedger will use higher confidence level to increase the amount of possible loss.^[35]

The application of (C)VaR, in particular the CVaR, in hedging with futures has not been widely investigated so far. Harris and Shen (2006) proposed a minimum-VaR hedging strategy that minimizes the historical simulation VaR of the hedge portfolio. Their study shows that although the minimum-variance hedging can reduce the standard deviation of portfolio returns, it tends to increase the portfolio skewness and kurtosis, and therefore, the reduction in VaR and CVaR from minimum-variance hedging is typically lower than the reduction in standard deviation.^[36] Hung et al. (2006) and Lee and Hung (2007) derived the optimal VaR hedge ratio with closed-form solution based on assumption that spot and futures returns are normal distributed. Chang (2010) selected a hedge ratio that meets the minimum VaR from the possible range of hedge ratios relying on numerical optimization techniques. Albrecht et al. (2011) provided analytical characterizations of minimum-(C)VaR hedging strategies for various parametric return models.

Following the framework of Albrecht et al. (2011), let Abbildung in dieser Leseprobe nicht enthalten denote probabilities, expectations and quantiles, which are conditional on available information set Abbildung in dieser Leseprobe nicht enthalten. Assume the distribution of Abbildung in dieser Leseprobe nicht enthalten given Abbildung in dieser Leseprobe nicht enthalten has a strictly positive density and both returns have bounded expectations for all Abbildung in dieser Leseprobe nicht enthalten.

Abbildung in dieser Leseprobe nicht enthalten-quantile of Abbildung in dieser Leseprobe nicht enthalten is defined for any confidence level Abbildung in dieser Leseprobe nicht enthalten as:

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If negative returns as (relative) losses are considered, VaR is defined as:

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CVaR is defined as:

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To stress the dependence of these risk measures on Abbildung in dieser Leseprobe nicht enthalten, VaR and CVaR can be expressed as:

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The objective functions can be expressed as:

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According to the Theorem 1 of Albrecht et al. (2011), the first order conditions are.^[37]

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Set the first order conditions equal to zero, the optimal hedge vector solve:

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Next section will present some specific assumptions regarding the return process.

3 Modeling Conditional Return Distribution

The return distribution is of utmost importance while hedging, because it describes the potential behavior of spot and futures prices in the future. Modeling the conditional distribution of asset price and return has been studied in the last three decades; it is a challenging task since asset returns exhibit several complicated features often referred as ‘stylized facts’. Asset returns seem to be leptokurtic with a large concentration of observations around the mean and have more outliers relative to the normal distribution. Volatility in asset returns does not appear evenly but high (low) volatility is expected to follow large (low) return changes of same sign, which is called volatility clustering. Volatility also has a tendency to rise more due to fall in asset prices as for rise in asset prices of the same magnitude, which is called leverage effect.^[38]

3.1 Relationship between Futures and Spot Prices

The futures price is the price specified in futures contract to deliver a specific quantity of a commodity at a specific future date, whereas the spot price is the cash price for immediate purchase and sale of the commodity, spot and futures prices converge at the maturity date of the future and a close relationship between the prices of futures contracts and spot prices should be expected. Brooks et al. (2002) stated, that “in the absence of transaction costs and market micro structure effects, or other impediments to their free operation, the efficient markets hypothesis and the absence of arbitrage opportunities imply that both markets react contemporaneously and identically to new information.”^[39] The exact nature of this relationship depends on many factors, for example seasonal effects, the nature of the commodity (storable or non-storable) and market expectations.^[40]

Numerous studies have focused on analyzing the cointegrating relationship between spot and futures prices.^[41] Before proceeding further, some fundamentals on stochastic processes will be presented below.

A series that is non-stationary, and only stationary after differencing a minimum of Abbildung in dieser Leseprobe nicht enthalten times is called ‘integrated of order Abbildung in dieser Leseprobe nicht enthalten’, denoted Abbildung in dieser Leseprobe nicht enthalten. An example of an integrated series of order 1 is a random walk.^[42]

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where Abbildung in dieser Leseprobe nicht enthalten denotes the drift and the error process Abbildung in dieser Leseprobe nicht enthalten is an independent and identically distributed (i.i.d.) stationary process. Alexander (1999) stated, that if financial markets are strongly efficient, the log prices are random walks and Abbildung in dieser Leseprobe nicht enthalten denotes the return at time Abbildung in dieser Leseprobe nicht enthalten.

Two integrated series are cointegrated if “there is a linear combination of these series that is stationary”. Cointegrated series have a common stochastic trend, they are tied together in the long run and the spreads are mean reverting, even though they might drift apart in the short run, i.e. if spot and futures prices are cointegrated, they would move together and cannot drift apart in a long run.^[43]

Granger (1986) and Engle and Granger (1987) stated that such a cointegration process should, in certain circumstances, be augmented to include an error correction term in each equation. Such model is called the vector error correction model (VECM) based on vector autoregressive process VAR model.

The vector autoregressive process Abbildung in dieser Leseprobe nicht enthalten is defined as

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where Abbildung in dieser Leseprobe nicht enthalten is a Abbildung in dieser Leseprobe nicht enthalten vector of variables that are Abbildung in dieser Leseprobe nicht enthalten, Abbildung in dieser Leseprobe nicht enthalten is a Abbildung in dieser Leseprobe nicht enthalten vector of innovations while Abbildung in dieser Leseprobe nicht enthalten through Abbildung in dieser Leseprobe nicht enthalten are Abbildung in dieser Leseprobe nicht enthalten coefficient matrices.

VAR can be transformed to first difference form by subtracting Abbildung in dieser Leseprobe nicht enthalten from both sides.^[44]

illustration not visible in this excerpt

where Abbildung in dieser Leseprobe nicht enthalten with Abbildung in dieser Leseprobe nicht enthalten and Abbildung in dieser Leseprobe nicht enthalten

The matrix Abbildung in dieser Leseprobe nicht enthalten determines the extent to which the system is cointegrated and Abbildung in dieser Leseprobe nicht enthalten is the appropriate number of lags. Information contained in matrices Abbildung in dieser Leseprobe nicht enthalten are short-term adjustment coefficients for the lagged differenced variables. Long run relationship between time series is indicated by the error correction term in Abbildung in dieser Leseprobe nicht enthalten. If the rank of the matrix is known, further conclusions regarding the number of cointegration relations can be made.^[45]

Empirical studies such as Kroner and Sultan (1993) indicate that “hedge ratios and measures of hedging performance may change sharply when this relationship is unduly ignored from the model specification” and “if spot and futures prices are cointegrated, omitting the equilibrium relationship will lead to misspecification problems by underestimating the true optimal hedge ratios.”^[46] This thesis will not enter into the debate from a theoretical viewpoint whether both markets must be cointegrated. However, this thesis will use cointegration techniques to investigate the existence of a long run relationship between spot and futures price series later in the Chapter 4.

3.2 GARCH Models

For estimating the optimal hedge ratio, the conditional variance and covariance of the spot and futures returns are required under the time-varying framework. The GARCH model and its enormous extensions and derivations have proved to be very useful in estimating jointly the conditional variances and covariances required for optimal hedge ratios.^[47]

The first family of models for the conditional distribution of asset returns was the Autoregressive Conditional Heteroscedastic (ARCH) introduced by Engle (1982).^[48]

Let Abbildung in dieser Leseprobe nicht enthalten be the price at time Abbildung in dieser Leseprobe nicht enthalten and assume conditional mean equations as

illustration not visible in this excerpt

where Abbildung in dieser Leseprobe nicht enthalten is constant drift term and Abbildung in dieser Leseprobe nicht enthalten refers all available information up to time Abbildung in dieser Leseprobe nicht enthalten. The Abbildung in dieser Leseprobe nicht enthalten represents the normal density distribution with zero mean, variance Abbildung in dieser Leseprobe nicht enthalten and degrees of freedom Abbildung in dieser Leseprobe nicht enthalten.

ARCH processes allow the current conditional variance to depend on squared errors in previous period and a constant. ARCH is specified as

[...]

^[1] See Hull (2007), p. 21.

^[2] See Parcell and Pierce (2000), p. 2.

^[3] For specification of futures, see Hull (2007), Chapter 2; McDonald (2009), pp. 138-139.

^[4] See Bhaduri and Durai (2008), p. 2.

^[5] See Lien and Tse (2002), p. 358.

^[6] See Harris and Shen (2004), p. 2.

^[7] See Chen et al. (2003) for different hedging strategies and objective functions.

^[8] When transactions and margin costs are ignored, the majority of the studies show that advanced econometric tools such as GARCH models improve the hedging performance over the naïve hedging strategy. Alexander et al. (2012), p. 1.

^[9] See Alizadeh et al. (2008), p. 1973.

^[10] This information set consists of the knowledge of all available values of the return series, and anything, which can be computed from these values, e.g. innovations, squared observations, etc. See Hurvich (n.d.), p. 2.

^[11] See Albrecht et al. (2011), p. 4.

^[12] See Acerbi and Tasche (2002); Rockafellar et al. (2004); Zhang and Rachev (2004); Szegö (2005).

^[13] See Föllmer and Schied (2008), p. 2.

^[14] All definitions are from McNeil et al. (2005).

^[15] The existence of a joint density implies the existence of marginal densities for all -dimensional marginal. However, the existence of a joint density is not necessarily implied by the existence of marginal densities. See McNeil et al. (2005).

^[16] See Markowitz (1952); Szegö (2005), p. 5.

^[17] See Rockafellar et al. (2004) for detailed discussion on deviation measures.

^[18] See Harris and Shen (2006), p. 370.

^[19] See Holton (2004) for discussion of the relationships between risk and uncertainty.

^[20] See Harris and Shen (2006), p. 370; McNeil et al. (2005) p. 43.

^[21] See Chung and Weon (2012), p. 2.

^[22] See Harris and Shen (2006), p. 371.

^[23] See Zhang and Rachev (2004), p. 12.

^[24] See Zhang and Rachev (2004), p. 12.

^[25] See McNeil et al. (2005), p. 243 for the proof.

^[26] See Embrechts (2000), p. 72.

^[27] See Acerbi and Tasche (2002) for numerical proofs of the total inadequacy of VaR as a risk measure.

^[28] See Szegö (2005), p. 1261; Rockafellar and Uryasev (2000), p. 22; Landsman and Valdez (2002), p. 56; for a discussion of numerical difficulties of VaR optimization, see Sarykalin et al. (2008); Rockafellar and Uryasev (2002).

^[29] See Cao et al. (2010), p. 781.

^[30] See Johnson (1960).

^[31] See Brooks et al. (2002), p. 335; Alizadeh and Nomikos (2004), p. 650.

^[32] See Ederington (1979) for hedging with T-Bill futures and Figlewski (1984) for hedging with stock indices futures.

^[33] See Lien and Tse (2002), p. 4; Harris and Shen (2002), p. 2.

^[34] The tests also have rejected the null hypothesis that time variation in optimal hedge ratios can be explained solely by deterministic seasonality and time to maturity effects.

^[35] See Lee and Hung (2007), p. 2405.

^[36] See Harris and Shen (2006), p. 371.

^[37] See Gourieroux et al. (2000), McNeil et al. (2005), Hong and Liu (2009) for quantile derivatives.

^[38] See Heracleous (2003), pp. 16-17.

^[39] See Brooks et al. (2002), p. 336.

^[40] See Baldi et al. (2011), p. 2.

^[41] See Engle and Granger (1987); Baillie and Myers (1991); Alexander (2004).

^[42] See Alexander (1999), p. 2041.

^[43] See Alexander (2008a), p. 246.

^[44] See Johansen (1988), p. 232; Sjö (2008), p. 14.

^[45] See Kühl (2007), p. 5; Johansen and Juselius (1990), p. 170.

^[46] See Alizadeh et al. (2008), p. 1971; Panagiotis (2011), p. 117; Brenner and Kroner (1995), p. 35.