Hedging Gasoil Commitments in Related Futures Markets
Master’s Thesis
in Advanced Economics and Finance1 by
Cecilie Viken and
Marianne Solem Thorsrud
November 16, 2013
Supervisors
Nina Lange2 and Kristian R. Miltersen3 Department of Finance Copenhagen Business School
1Number of Characters (Pages): 269,133 (118)
2e-mail: nl.fi@cbs.dk
3e-mail: krm.fi@cbs.dk
Cecilie Viken and Marianne Solem Thorsrud
This thesis was written using the typewriting program LATEX, 12pt.
The German gasoil market does not have an existent liquid futures market. In order to hedge risks in this market, other correlated markets must be used for hedging.
We look at proxy hedges using both Brent crude oil futures and ARA gasoil futures, when trying to reduce spot price risks in the formerly mentioned German gasoil market.
A 3-factor model is introduced, where the spot price and the convenience yield are modeled as stochastic, and the log spread between the illiquid spot price and the futures price used to hedge is modeled as a mean-reverting OU process. This model is estimated through a Kalman filtering method. When modeling the log spread as an OU-process, we implicitly assume a stationary distribution of the log spread and thereby indirectly account for a long-term cointegration relationship between the spot price being hedged and the futures price used to hedge.
Different hedging strategies are tested on the prices that are simulated through the 3-factor model. These hedging strategies include a regression based minimum variance hedge, a parameter-based minimum variance hedge, log and quadratic util- ity hedges and a na¨ıve hedge. These hedges are then compared to the strategy without any hedge. Ederington’s measure of effectiveness and Value-at-Risk are computed to check the quality of the results. We finally test our hedging strategies on a specific case, where DONG Energy is committed to buy German gasoil in the spot market. The hedges are tested on a contract price based on the spot price of German gasoil.
Our results show great effectiveness in both markets, but ARA gasoil futures are more effective in reducing the risks coming from the German gasoil market. We also find that a static hedging strategy is more effective than both static rolling and semi- dynamic strategies. This also holds for the case applied to DONG Energy, a large Danish energy company hedging against price risk in the German gasoil market.
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This master’s thesis was written in partial fulfilment of obtaining the degree M.Sc.
in Advanced Economics and Finance (cand.oecon) at Copenhagen Business School.
First and foremost, we would like to express our gratitude to our supervisors, professor Kristian R. Miltersen and PhD Graduate at DONG Energy Nina Lange.
You have both been of great inspiration, making us eager to learn about energy mar- kets and thereof writing this thesis. You have helped us with both theoretical and practical obstacles and guided us in the right direction, something for which we are truly thankful. Second, we would like to thank DONG Energy for the information and data provided to us. Lastly, we would like to thank our families and friends for their patience and support when writing this thesis.
Cecilie Viken4
Marianne Solem Thorsrud5
Oslo, November 2013
4e-mail: viken cecilie@hotmail.com
5e-mail: marianne.thorsrud@gmail.com
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Executive Summary i
Aknowledgements ii
1 Introduction 1
1.1 Introducing the Problem in Question . . . 2
1.2 Organization of the Thesis . . . 2
2 Overview of Energy Markets 4 2.1 Introduction to Energy Markets . . . 4
2.2 Characteristics of Energy Prices . . . 6
2.3 The Crude Oil Market . . . 8
2.3.1 The Value of Crude Oil . . . 9
2.3.2 Trading North Sea Oil . . . 10
2.4 The Gasoil Market . . . 12
2.4.1 Refining Crude Oil . . . 12
2.4.2 The European Gasoil Market . . . 13
3 Review of Futures Markets and Hedging 15 3.1 Futures Markets . . . 15
3.1.1 Forward and Futures Contracts . . . 15
3.1.2 Equilibrium in the Futures Market . . . 16
3.1.3 Liquidity . . . 17
3.1.4 Futures Prices . . . 19
3.2 Risk Management of Commodities . . . 22
3.2.1 Exchange-Traded Products vs. OTC . . . 22
3.2.2 Different Types of Risk . . . 23 iii
3.2.3 Hedging with Futures . . . 25
3.3 Optimal Futures Hedge Ratios . . . 27
3.3.1 Minimum Variance Hedge Ratio . . . 29
3.3.2 Utility-Based Hedge Ratios . . . 32
3.3.3 Dynamic Hedge Ratios . . . 38
3.4 Proxy Hedging . . . 43
3.4.1 Liquidity versus Correlation . . . 44
3.4.2 Long-Term Relationship . . . 45
3.5 Short Summary . . . 46
4 Modeling of Prices 47 4.1 Introduction . . . 47
4.2 Stochastic Processes . . . 47
4.2.1 Wiener Process . . . 49
4.2.2 Generalized Wiener Process . . . 50
4.2.3 Itˆo Processes and Itˆo’s Lemma . . . 52
4.3 Spot Price Models . . . 53
4.3.1 Geometric Brownian Motion . . . 53
4.3.2 The 1-factor Model . . . 55
4.3.3 Gibson-Schwartz’ 2-factor Model . . . 57
4.4 Relationship between Spot and Futures Prices . . . 58
4.4.1 Risk Neutral Probabilities . . . 58
4.4.2 Pricing of Futures Contracts . . . 60
5 The 3-factor Spot Price Model 63 5.1 Rationale . . . 63
5.2 Data . . . 64
5.3 The 3-Factor Model . . . 70
5.3.1 General Setup . . . 70
5.3.2 Cholesky Decomposition of Correlations . . . 71
5.3.3 System of Equations to be Simulated . . . 73
5.4 State-Space Models and the Kalman Filter . . . 75
5.5 Estimation and Results . . . 78
5.6 Simulation of Prices . . . 81
5.7 Discussion . . . 83
6 Proxy Hedging 85
6.1 General Setup . . . 85
6.1.1 Minimum Variance Hedge . . . 86
6.1.2 Mean-Variance Hedge . . . 87
6.1.3 Na¨ıve Hedge . . . 90
6.1.4 Effectiveness . . . 90
6.2 Hedging Strategies . . . 91
6.2.1 Static Hedge . . . 92
6.2.2 Static Rolling Hedge . . . 95
6.2.3 Semi-Dynamic Hedge . . . 98
6.3 Discussion . . . 110
7 Case: DONG Energy buying Gas 113 7.1 Continental European Term Contracts . . . 113
7.2 Contract Setup . . . 114
7.3 Results . . . 116
8 Conclusion 120 Appendix A Figures, Tables and Mathematical Proofs 122 A.1 Figures . . . 122
A.2 Tables . . . 125
A.3 Mathematical Proofs . . . 135
Appendix B R Codes 137 B.1 Data Description . . . 137
B.2 Simulations of Wiener Processes . . . 141
B.3 Stationarity Tests of Data . . . 143
B.4 Kalman Filter - FKF Package . . . 145
B.5 GARCH(1,1)-M Modeling . . . 151
Bibliography 158
Introduction
The motivation behind taking on a thorough study of hedging in energy markets stems from the Energy Markets and Real Options course taught by Kristian Mil- tersen and Nina Lange, which is a progressive course in relation to the cand.oecon program at Copenhagen Business School. As a part of this course we were allowed to conduct a case study for DONG Energy, one of Denmark’s largest energy com- panies. This case opened our eyes to financial modeling of commodity prices and introduced us to stochastic processes and stochastic models of doing so. In the wake of this, it became more or less obvious for us that it was within this field we wanted write our thesis, which we knew would be both educational and challenging. Not to mention the possibility to work under the supervision of Kristian Miltersen and Nina Lange, and receiving valuable input from DONG Energy.
With the help of our supervisors we were able to narrow the field down to an interesting topic within financial modeling in commodity markets, namely proxy hedging in energy markets. After timely research of past analyses of this topic, we chose to base the thesis on mainly two papers written by Ankirchner, Dimitroff, Heyne, and Pigorsch (2012) and Bertus, Godbey, and Hilliard (2009). Proxy hedg- ing in energy markets is a rather new research area, and is becoming increasingly important as different markets are still under development and for which there does not exist liquid derivatives markets. For this reason we hope that with our empirical application of different ways of hedging in energy markets, we are able to contribute to past research and spur interest within this particular domain.
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1.1 Introducing the Problem in Question
In this thesis we explore the problem a large energy company faces when it wishes to hedge a commitment to buy or sell in the spot market at certain time period when there does not exist a liquid futures market, and must then turn to other futures markets to hedge.
The market for oil with its refined products has become more and more dereg- ulated and highly liquid financial and physical markets have emerged with highly volatile prices. This has resulted in a significant exposure to financial risks and thus the importance for effective risk management has steadily increased. However, for most refined products the market is divided into geographical units, for which there do not exist liquid derivatives to hedge with.
The focus in this thesis is in a sense twofold. The main goal is to test different strategies a company can undertake when hedging exposures in the German gasoil market by using futures contracts on Brent crude oil and ARA gasoil. The different strategies are based on past research and entails static, rolling and semi-dynamic hedges. In addition, how much risk a company would wish to undertake is also considered, whether it would be minimizing the variance of cash flows, exploring the trade-off between risk and return, assuming a one-to-one hedge or not hedging at all.
The second goal of the thesis is to find a good stochastic model for modeling both the underlying spot prices of the futures one is hedging with and also the spread between spot prices one is seeking to hedge and the futures prices. The stochastic model is based on price models from financial modeling theory, where it in reality combines two well-established models by introducing correlation parameters so as to take into account both the correlation between convenience yield and spot, and also the correlation between spot and spread, resulting in a 3-factor model. This 3-factor model was introduced by Bertus, Godbey, and Hilliard (2009), and is the foundation for our analysis.
1.2 Organization of the Thesis
The remainder of the thesis is structured as follows. Chapter 2 will give an introduc- tion to energy markets, with a focus on the European crude oil and gasoil markets.
It also presents the main drivers and characteristics of energy prices. In Chapter 3 we give a review of futures and forward markets, their prices and liquidity. We also introduce different ways of hedging with futures contracts, which will be the basis of the further analysis. How to model prices is introduced in Chapter 4, where we present different stochastic processes and models. Chapter 4 leads the way to Chap- ter 5 where we define our 3-factor model with its underlying assumptions. We show how prices and spreads are simulated as well as how the parameters in the model are estimated using the Kalman filter. Moreover, the resulting parameters and a discussion of the model is provided. Chapter 6 focuses on the different ways of hedg- ing an exposure in the German gasoil market with both Brent crude oil and ARA gasoil futures. For each hedging strategy the results are presented and discussed.
In Chapter 7, we extend our thesis to a realistic case provided by DONG Energy, where we test our findings to a practical problem. Finally, Chapter 8 summarizes and concludes our thesis.
Overview of Energy Markets
In this chapter we give an overview of energy markets. Although theterm “energy markets” covers a variety of commodities, such as fuels, electricity, weather and emissions, our focus will initially be on fuels. We provide a short introduction to oil and gas markets, and then continue concentrating on the crude oil market and its refined products in Europe, as they are the most relevant for our analysis.
This chapter is based on Schofield (2007), Burger, Graeber, and Schindlmayr (2008), Eydeland and Wolyniec (2003), Geman (2005) and current market informa- tion from the Intercontinental Exchange (ICE).
2.1 Introduction to Energy Markets
Before the liberalization of the energy sector, oil and gas markets, amongst others, were heavily regulated. Regulation in itself was a response to the fear of monopoly power stemming from activities in energy markets being controlled by only one entity (Eydeland and Wolyniec, 2003). With regulated markets, participants were allowed to base prices on cost, thus transferring all cost to the end user, which meant that they were not exposed to significant financial risks (Burger, Graeber, and Schindlmayr, 2008). The inefficiency of this situation led to the liberalization of the energy sector and thus required energy companies and utilites to come up with new ways of doing buisness. Competition accruded which led to end user prices now being market based instead of cost based. This also meant a significant exposure to financial risks; both gas and oil are highly volatile commodities, and consumer demand is uncertain and often seasonal. Thus the need of efficient risk management
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is becoming increasingly important. In fact, data analysis suggests that energy prices are much more volatile than other asset types such as interest rates, foreign exchange rates and equity prices (Eydeland and Wolyniec, 2003).
Energy became a tradable commodity during the 1980s with the development of the oil market. From this and 20 years later the oil market has become the greatest commodity market in the world, with both a very large physical market and a large financial market (Geman, 2005). In addition, as a result of a variety of deregultations in the U.S., a liquid and competitive natural gas market emerged in the early 1990s. However, in Europe, this only exists in the U.K. at this point (Burger, Graeber, and Schindlmayr, 2008).
Different products are available when trading energy; spot transactions, for- wards, futures, swaps and options. Forwards is referred to as bilateral agreements and are traded over-the-counter (OTC), whereas futures are standardized agree- ments and are concluded at a commodity exchange. The most significant exchanges from a global perspective is theNew York Mercantile Exchange (NYMEX) and the Intercontinental Exchange (ICE).
Being the world’s largest physical commodity futures exchange, NYMEX offers a variety of products such as futures and options contracts for energy and metals.
Introduced in 1983 and 1990, the NYMEX light sweet crude oil futures contract and the NYMEX Henry Hub natural gas futures contracts, respectively, are the most popular benchmarks in the United States (Burger, Graeber, and Schindlmayr, 2008).
When founded in 2000, the main objective for the ICE was to offer an electronic platform for OTC energy commodity trades. However, by acquiring the Interna- tional Petroleum Exchange (IPE) in 2001, the ICE expanded into futures. Today the IPE is known as ICE Futures Europe. Although the ICE is a relatively new exchange, the IPE has existed since 1980. The first contract offered by the IPE was gasoil futures, but the Brent crude futures contract followed in 1988. Today, the ICE offers, amongst other contracts, the benchmarks Brent crude oil, West Texas Intermediate (WTI) crude oil and gasoil futures contracts (the Intercontinental Ex- change)1.
1www.theice.com/publicdocs/ICE Crude Oil.pdf
2.2 Characteristics of Energy Prices
In commodity markets, as with any other financial market, we separate between spot and futures markets. When trading in the spot market, delivery takes places either right away or with a lag, depending on the commodity being transacted. As with for example gas, there is a lag of one day (day-ahead market) before the buyer actually receives the gas due to technical constraints of delivery. The commodity is then traded at spot price (or day-ahead price).
Many commodity markets have today become integrated, and energy companies trade in the spot market in order to face increased demand, previously covered by own supply (Fusaro, 1998). The past decades, there has been a steady increase in the volatility of commodity prices. This mainly stems from larger variation in supply and demand (Geman, 2005). Weather, income levels, demand for cars, raw material prices etc. are only some of the reasons why we observe large variations in supply and demand (Pindyck, 2001). Even though prices fluctuate much, they do so partly predictably given theseasonal demand of many commodities. A typical example of seasonal demand is the demand for natural gas, which is often used for heating, thus demand typically rises during winter months and then falls during summer months.
Yet, since the level of volatility also might vary over time, there is to a large degree uncertainty about where future prices are headed (Pindyck, 2001). In addition, one can often see suddenspikes orjumps in prices, resulting from unforeseen events such as outages of electricity - another volatile energy market.
Common to many commodities is the issue of non-storability, which makes prices vary even more with supply and demand. This is also one of the main reasons for different geographic markets in some commodities. However, when talking about storable commodities, as for example oil, how much is stored by producers conveys essential information about how they believe supply and demand will evolve. If demand is expected to go up, additional inventories could meet this extra demand and keep the market balanced pricewise, or as in a normal competitive market; the inventories could be kept fixed, and as production then must increase, so will the prices until demand is back to normal. The second scenario is known to be common in most commodity markets; prices tend to revert back their “normal” level - so calledmean reversion of prices.
This concept of whether to store the commodity or buy extra from elsewhere
when needed is one of the specialties of commodity markets compared to the financial markets. Producers and consumers must always evaluate whether it is beneficial to have inventories - is it better to sell what we have stored now in the spot market, or should we continue building the inventories up in case of supply shortages or spikes in demand that might be encountered at some future date (Pindyck, 2001)? The (net) convenience yield of a given commodity is defined as the (net) flow of services (per unit time and per monetary unit of the commodity) that accrues to a holder of the physical commodity, but not to a holder of a contract for future delivery of the same commodity (Brennan, 1991). The net convenience yield is as a result the difference between the gross convenience and the cost-of-carry of holding the commodity (Miltersen, 2003).
As with the price volatility, the convenience yield might vary over time. The spot price is negatively correlated to inventories, because when spot prices go up, one would want to sell off inventories. When one reduces inventories, there is more scarcity for future unpredicted events, and the convenience yield would normally go up (it becomes valuable to hold). Therefore one sees a negative correlation between the convenience yield and the inventory level, and ergo a positive correlation between the convenience yield and the spot price as they are both negatively correlated to inventory levels. Knowing that spot prices are as volatile as they are, it makes sense that the convenience yield in itself also varies over time.
The inventory market is thus related to the spot market, and production does not necessarily have to be equal to demand (or consumption) at a given point in time. The spot market is, as a consequence, the relationship between the spot price and the net demand, which is defined by Pindyck (2001) as the difference between production and demand (or consumption). The spot prices of storable commodities are thus dependent on both of these markets, and events not accounted for can have a large impact on prices making them vary much and jump. In summary, there are some features special to commodity prices that need to be taken into account when modeling and predicting prices.
Furthermore, it is important for any market participant to be aware of the cur- rent term structure of the prices, that is, whether the market is in contango or backwardation. The market will experiencecontango when forward prices are higher than spot price, andbackwardation when they are lower than spot prices (Schofield, 2007). For instance crude oil is more prone to backwardation. The reason is that
crude oil is relatively expensive to store, which discourages storage. This means that when there is an increase in demand, current stock will possibly not be able to meet this demand. This creates a time lag between spot and forward price, and thus spot prices rise relative to forward prices and possibly pushing the market into backwardation.
Typically, backwardated markets are characterized by a scarcity of the commod- ity, low inventories, volatile prices due to low inventories and a strongly rising price.
Contango markets, on the other hand, are characterized by an excess of the com- modity, high levels of inventory, relative price stability and general price weakness (Schofield, 2007).
2.3 The Crude Oil Market
Crude oil is the world’s largest primary energy source and according to the BP Statistical Review of World Energy (2013) it covers approximately 33% of world- wide energy consumption and 30% of European consumption. Although the major- ity of consumption is based in North America, Europe and Asia, the majority of production is located in developing or transition countries (Burger, Graeber, and Schindlmayr, 2008).
The fact that oil is a finite resource has had a significant effect on the supply side. The fear of depletion has led producers to invest more and more resources to discover new reserves, while they bring to the market smaller and smaller quantities of oil. Furthermore the important effect crude oil has on the global economy makes it subject to political regulations and interventions. Thus crude oil prices have always been highly influenced by political and geopolitical events.
For instance, as seen in Figure 2.12, downturns in the global economy in the early 1980s, in 1998 and 2008 resulted in sharp decreases in the oil price as well.
One of the main drivers of the oil price is demand, and this together with the scarcity of resources, political events and random shocks results in prices being highly volatile; “despite more than 150 years of effort, the next person to forecast crude oil prices successfully for any sustained period of time will be the first” (Intercontinental Exchange)3.
2Figure taken from Burger, Graeber, and Schindlmayr (2008) and www.econstats.com
3www.theice.com/publicdocs/ICE Crude Oil.pdf
Figure 2.1: Historical Brent Spot Prices
2.3.1 The Value of Crude Oil
There exists an array of different types of crude oil, where the value is determined based on each the different types. The name “crude oil” only specifies the state of oils before they are refined (Schofield, 2007). Each of the different types has different characteristics and each is attractive for different reasons, determined by their gravity, viscosity and sulphur content (Eydeland and Wolyniec, 2003). Based on these characteristics crude oil is often divided betweenheavy and light depending on their gravity and viscosity, and between sweet and sour crude oil depending on the sulphur content. For instance, the benchmarks, WTI and Brent, are typical light and sweet crude oils, meaning they have a lower gravity and viscosity values, and have a sulphur content below 0,5% (Eydeland and Wolyniec, 2003). Thus the different characteristics of crude oils determine the value of the oil, which essentially means that value lays in what kind of product can be refined from the crude oil.
When oil is transported, it is done both in ships and pipelines, however across the international market it is exclusively transported in ships (Geman, 2005). For this reason the price of oil is most often expressed as FOB (Free On Board), which refers to the point of loading and makes it possible to compare different crude oil prices around the world (Schofield, 2007). Another way of pricing is CIF (Cost, Insurance and Freight), which gives an indication of the cost of delivering crude oil.
Furthermore the price of oil is mainly quoted in U.S. dollars per barrel, and is often shortened to USD per bbl.
With so many different types of oil and prices, it can be difficult to understand
how a single market for oil has developed. The reason is, as mentioned in the introduction of this chapter, that the industry has focused on a small number of references, or benchmarks, for which other types of oil are priced after as differentials (Geman, 2005). The two major benchmarks are Brent (North West Europe origin) and West Texas Intermediate (USA origin).
Although a large part of oil pricing is done by setting a differential to one of the benchmarks, other methods also exists (Schofield, 2007). These methods entail governments setting an official selling price, two parties negotiate a fixed price, companies that specialize in gathering and publishing reference prices set afloating price, prices are set relative to futures prices and two counterparties agree to do a physical trade of oil and define the price as anexchange for physicals.
Thus, when pricing a delivery contract, the nature of the price and what is actually going to be delivered must be specified in the contract. Typical details included are the price (USD per bbl), whether the contract is priced off an index or marker crude, any differential that should be applied to the price, the quality of the crude oil, time and place of delivery and whether the price is FOB or CIF.
2.3.2 Trading North Sea Oil
As the focus in this paper will be on the European market for oil, we now proceed by defining the different ways of trading the crude oil attached to North Western Europe, namely Brent.
For a long time Brent blend, which is a mixture of crude oils from many different wells, was the representative price of North Sea oil. However, as a result of the decline in oil production the representative price was extended to include activity in four crudes: Brent blend, Forties, Oseberg andEkofisk (BFOE). The three last ones were chosen because they are considered to have similar characteristics to those of Brent Blend (Schofield, 2007). The new price now ensured that it reflected actual market activity. Brent is used as a primary hedging tool for many different reasons such as its accessibility and reach as a seaborne crude, production, and adaption to changing global economies in the oil market, stability and geographic location.
These reasons also explain why it is very common to use it as a benchmark.
In practice the price of Brent is determined in five market divisions (Geman, 2005). The first is the spot market, or more commonly known as dated Brent.
When talking about “spot” in relation to oil markets, it has a somewhat different
meaning. Normally “spot” would be considered to be immediate delivery, yet oil is rarely traded for delivery in less than 10 days. For this reason dated Brent is considered as the spot market and is in practice a cargo loading within the next 10-21 days (Schofield, 2007).
The second division is the forward market. The Brent forward market reaches beyond 21 days and locks in the price at the time of contracting for delivery in the future (Schofield, 2007). When entering a Brent forward contract the exact day for when the oil is loaded is not stated, but rather the month of loading. Thus the price reflects the value of the cargo for physical delivery within this month. Furthermore the Brent forward market is often referred to as the 21-day market, since buyers are notified 21 days in advance of the dates for loading.
The third division is the market for CFDs (Contracts For Differences). CFDs are purely financial swaps used for hedging basis risk (Geman, 2005) and represent the difference between the current second month forward quote and Dated Brent quotations over a given loading week (Schofield, 2007).
The fourth division is the Brent futures market, where crude oil is traded with an exchange-traded futures contract. The Brent futures contract is highly attractive because of its deep liquidity and far reaching forward curve (Schofield, 2007). It is mainly traded on the ICE and is physically deliverable with an option to cash settle at expiry. As for the Brent forwards, Brent futures only specify the month of loading and not the exact date. Although the contracts are available for maturities up to 6 years, liquidity tends to decrease as maturity increases, just as for other futures contracts for energy commodities. Typically only the next few months are traded liquidly.
For longer term maturities, there is a liquid OTC market for Brent swaps, which represents the fifth division. The underlying are the first nearby ICE Futures Europe contracts, and the maturity for the swap extends to 10 years forward (Geman, 2005).
As a last note, it is worth mentioning that first and foremost crude oil is bought and sold to balance supply and demand. In general, trading opportunities arise when a producer has available crude oil to sell, when demand exists within the supply chain for a particular crude oil, when there is demand for a particular quality of crude oil or a trader has discovered an opportunity to make money.
Typically, participants will be oil companies, producers, refineries and financial institutions. After a while, each participant will end up with a portfolio of different
types of crude oils, of different qualities and grades, that are priced differently and with different times to maturity.
From this, a second motive for trading crude oil arises, which is to mitigate the market risks that arise from selling and buying crude oil. As with financial stocks, these risks can be managed by using derivatives, which we will show later in this paper.
A third motive for trading is speculation, that is, market participants that wish to make money from an anticipated move in some market variable. This is done by trading on the ups and downs in the spot price, on grade differentials, on the shape of the forward curve, price volatility and the relationship between markets.
2.4 The Gasoil Market
For our analysis’ sake, it is worth mentioning the European gasoil market. Gasoil is one of the most important refined oil products, especially in Europe, and has a fairly liquid futures market.
2.4.1 Refining Crude Oil
Refineries convert crude oils into various products, where each refinery is different to the next one and is differentiated by its ability to create a high-value end product.
The most important products aregasoline, jet fuel, diesel fuel and asphalt (Burger, Graeber, and Schindlmayr, 2008). Due to the refining process, prices of different products are often tightly related, and can be expressed in terms of price spreads against crude oils. The lighter and more valuable products have higher spreads against crude oils than heavier products.
Refineries are concerned with maximizing their revenue, referred to as thegross production value. This value can be used to determine the value of crude oil, or more precisely the netback value of crude oil (Schofield, 2007). The netback value is the total value of all the refined products after having subtracted all production costs (e.g. transportation, insurance, operational costs and financing). This value can then be compared to the current market value of oil and thus one can determine whether it is more economical to sell or buy oil.
2.4.2 The European Gasoil Market
In Europe, the refining industry is defined by geography and is divided into ARA (Amsterdam-Rotterdam-Antwerp) andMediterranean (Genoa). ARA is the obvious center for refined products as it has a dense network of river canals, deep-water ports and other transportation infrastructure (Burger, Graeber, and Schindlmayr, 2008).
The most important refined product in Europe is gasoil, which is what is called a middle distillate, heavier than gasoline, but lighter than heavy fuel oils. Gasoil is mostly used for domestic heating and transportation (diesel). One important characteristic of gasoil is that it is a less volatile substance than for instance gasoline, which simplifies storage and transportation (Intercontinental Exchange)4.
Although factors in the global crude oil market plays a significant role, the most important determinant of the price of gasoil is, similar to crude oil, supply and de- mand. However, due to refining capacities, supply does not always meet demand5. Some refineries have excess capacity, producing more than local demand, and others lower capacity thus not meeting demand. This may result in higher transaction costs or high discounts and premiums when trading. The reason for this asym- metry between refining capacity and consumer demand is that expanding capacity takes much more than just increasing production; the whole infrastructure, includ- ing pipelines, tanks, power supplies etc, must also be expanded (Intercontinental Exchange)6. Furthermore, this constrained capacity leads to a highly volatile gasoil price, especially compared to the crude oil price (the spread between the crude oil price and the gasoil price is referred to as the crack spread). A sudden increase in demand, resulting from for instance cold weather or sudden supply shocks, may spur a sudden jump in the price. For this reason the price of gasoil is seen to be highly seasonal.
As mentioned in the beginning of this chapter, one of the first contracts intro- duced by the IPE (now the ICE Futures Europe) was gasoil futures. Today, the price of ICE gasoil futures serves as a benchmark for many markets such as in Rus- sia, the Middle East, Asia and also other markets in Europe. Thus the ICE gasoil futures contract has become a very important contract for traders trading in energy markets, where the key specifications of the contract are summarized in Figure 2.2
4www.theice.com/publicdocs/ICE Gasoil Brochure.pdf
5www.refinerlink.com/blog/How Traders Should Buy and Sell Refinery Gasoil
6www.theice.com/publicdocs/ICE Gasoil Brochure.pdf
below (Intercontinental Exchange)7:
Figure 2.2: Example of Gasoil Futures Contract Characteristics
7www.theice.com/publicdocs/ICE Gasoil Brochure.pdf
Review of Futures Markets and Hedging
As mentioned in the previous chapter, the need for risk management when it comes to volatile energy prices has become increasingly important. For this reason, a logical next step in this thesis is to focus on mitigating risk in commodity markets.
The following chapter is mainly based on the books by Geman (2005), Fusaro (1998), Eydeland and Wolyniec (2003) and Duffie (1989) unless otherwise stated, where the three former are primers on commodities, their derivatives and how to manage commodity risk and the latter on futures markets.
3.1 Futures Markets
The market for financial instruments rose as a response to the volatile commodity markets (Pindyck, 2001). Futures and forward contracts are among the most widely used hedging tools today, and exactly how these work in mitigating or reducing risk will be explained in Section 3.2. The goal of this section is to cover the basics of forward and futures markets: how they function and how the forward and futures prices are determined.
3.1.1 Forward and Futures Contracts
Following the definition of Duffie (1989), “a forward (or futures) contract is an agreement between two parties to make a particular exchange at a particular future date”. The buyer of the contract will thus at a future date receive the commodity
15
for a predetermined forward (futures) price that he or she pays at delivery. The time frame of the contract may vary from one month to ten years depending on the commodity in question. For example (Geman, 2005), in a forward agreement, party A is the seller of the forward contract, and is thus obliged to deliver at some future date an underlying commodity. Party B is the buyer of the contract, and is thus obliged to sell the underlying at that same time. We say that party A is short forward, who wants to hedge against lower prices, and B is long forward, who wants to hedge against a rise in prices. At maturity, they clear their forward positions.
A futures contract functions in the same way as a forward contract, but the main difference between the two is that a futures contract is marked-to-market every day during the contract’s life through a Futures exchange. It is also standardized in its terms of agreement. In a futures contract, the agreeing parties adjust their positions daily as the price of the underlying commodity changes.
When trading in the futures market, there are some risk factors that should be taken into account (Eydeland and Wolyniec, 2003). Basis risk arises when, at expiration, the futures price is not exactly equal to the underlying spot price.
Liquidity risk is the inability to enter into a futures position at the right time, at the quoted price and at the right size. Credit risk refers to the inability of counterparties meeting their cash obligations (or margin requirements).
The futures contract is less risky than the forward contract, as the clearinghouse eliminates credit risk because of the margin deposits - one initial deposit and daily deposits made throughout until maturity of the contract. If the risk-free interest rate is assumed constant, the forward and futures contracts are equal. This is because when the interest rate is constant, the discounted forward contract will be no different than the non-discounted futures contract. A futures contract will differ from the forward when interest rates are assumed to vary over time, since then the daily settlements will have a different impact on the prices (Eydeland and Wolyniec, 2003).
From here on out, we will explicitly use the term futures prices or futures con- tracts, simply because we later on will assume a constant interest rate.
3.1.2 Equilibrium in the Futures Market
In the economic theory of competitive markets, one determines the market price in equilibrium, where supply equals demand. Each market participant decides on
his or her optimal position given the market price. This also holds for the futures market, where, in equilibrium, the total of all short positions must equal the total of all long positions. As Duffie (1989) asks, how well does the competitive equilibrium fit into the futures market setting? There is no initial supply of futures contracts, so the supply in equilibrium is set to zero. The demand consists of market participants wanting to go both short (sell) and long (buy) in futures, and in equilibrium, they sum up to zero. Those being short contribute negatively to demand, whilst those being long contribute positively. Individual demands are thus different than zero, but their sum should always equal zero. If the futures price is very high for example, then demand is more likely to be negative (e.g. one would want to sell futures) and vice versa if futures prices are likely to be low. In sum, for every short position there must be an offsetting long position.
Most often, futures contracts are not held until maturity, but they are settled with cash beforehand. This is also one of the reasons why many investors trade in futures, since there is not an issue of whether to actually buy (or sell) the commodity or not - one can just settle financially without any physical delivery. Anyone who wishes could take short or long positions in commodities, without having to receive (or deliver) the actual commodity. The futures market consists therefore of both hedgers and speculators.
3.1.3 Liquidity
Liquidity can be measured as volume sold and bought, or defined another way; the size of the trade it takes to affect the market (Geman, 2005). Not surprisingly, oil contracts are by far the most traded in the U.S., but gas contracts follow quite closely.
There also seems to be higher liquidity for the contracts maturing sooner rather than later. Today, most futures contracts are relatively liquid where they exist, and especially so for energy commodities (Geman, 2005). Open interest is another measure of how much trading activity there is, and the term refers specifically to how many contracts are outstanding at a given point in time, i.e. the total number of short and long positions in the market.
Figure 3.1 shows exactly how much the volume and open interest has increased during the past years for Brent crude oil futures. The Brent market is the oldest forward market in oil, and the development of a successful forward market lays the grounds for a successful futures market in the same (or similar) commodity.
(a) Open Interest Brent Futures
(b) Volume Brent Futures
Figure 3.1: Liquidity of Brent crude oil
For ARA (Amsterdam, Rotterdam, Antwerp) gasoil, both open interest and volume has also increased during the past years even though the scale is smaller, as can be seen from Figure 3.2. Common for both Brent crude oil and ARA gasoil is that the longer the maturity of the futures contract, the less liquid it is. In addition, liquidity - i.e. both volume and open interest - has increased steadily throughout the past years from 2005.
(a) Open Interest ARA gasoil Futures
(b) Volume ARA gasoil Futures
Figure 3.2: Liquidity of ARA gasoil Futures
3.1.4 Futures Prices
One implication of liquid futures markets is price transparency. The possibility of arbitrage profits is therefore lowered, and the futures price is a way of telling how the price of the underlying commodity will evolve. In illiquid markets on the other hand, there might be a possibility of manipulating the prices of commodities, given the definition above of liquidity.
Forward or futures prices tend to become less volatile as their time to matu- rity increases, or equivalently; more volatile as expiration approaches. This is the so-calledSamuelson Effect, which Paul A. Samuelson discovered in 1965. This argu-
ment stems from the fact that new information always should be taken into account.
The estimate of the variance in futures prices is made given the information avail- able at that point in time, the so called conditional variance estimate (Duffie, 1989).
The hypothesis made by Samuelson says that information is received more rapidly as expiration approaches, and that this would imply higher volatility in prices as expi- ration approaches. As Duffie (1989) further explains, this might only be a symptom of seasonal prices, or liquidity.
The forward curve shows different futures prices (y-values) as a function of their time to maturity (x-values) seen from a fixed point in time. In short, it is a term structure of forward or futures prices observed in the market at a given point in time (Eydeland and Wolyniec, 2003). It serves as a good tool for those interested in hedging and speculating, given its indications of where future prices are headed.
In a more general statement, the forward or futures curve gathers the expectations we have about future spot prices through variable supply and demand, seasonality etc., and it is therefore important that this term structure is replicated as closely as possible when modeling spot and futures prices.
As mentioned in Section 3.1.1, the convenience yield measures the net benefit from holding the commodity versus being long in a futures contract. The difference between the spot price today and the futures price in perfect markets is due to this convenience yield and normal discounting. The reason for mentioning perfect markets is because in such markets, forward or futures prices will converge to the spot prices at maturity. When markets are imperfect, this is not necessarily the case, and there will be some risk involved in terms of how the spot and futures prices move together. This is the so called basis risk, mentioned earlier.
So do futures prices perfectly predict the future spot prices? In other words, is the expected future spot price equal to today’s futures price? Below is a relationship that is often seen in books about commodities and energy prices. If the futures price today, at timet, of a commodity that will be sold / received at time T, equals today’s expected spot price under Q1 at time T, then we can predict future spot prices through the futures price.
Ft(T) =EtQ[ST] =Ste(r−δ)(T−t) when t ≤T
1More information about the risk-neutral Q-measure will be given in Chapter 4, Section 4.4
If the above relationship holds, futures prices are martingales and they can perfectly predict the future spot prices. In this relationship, r is the risk-free interest rate and δ is the convenience yield. It is also known as the convergence assumption (Eydeland and Wolyniec, 2003), where it is assumed that at maturity, the forward or futures price converges to the spot price ST underlying the futures contract. In the real world, this expected value of ST would probably be based on an analysis of historical data or some forecasting method. Physical world expectations are not the same as risk-neutral expectations (Eydeland and Wolyniec, 2003).
Relevant to the discussion about martingale futures prices is the topic of auto- correlation, or serial correlation. If prices are serially correlated, then past prices are useful in predicting future prices. Autocorrelation is important to consider when doing empirical analyses with time series, as presence of it would bias the ordinary least squares (OLS) estimates. Furthermore, if futures prices are martingales, or more correctly the futures price process
Fi =F1, F2, . . . , Fn with i∈(1, n) is a martingale then the futures prices show no autocorrelation.
If the forward or futures price is lower than the spot price today, we observe backwardation, and the convenience yield is lower than the risk-free rate (under the Q-measure and assuming a constant interest rate). It then makes sense to sell off the stock you have, and instead take up a long position in forwards or futures.
Backwardation is normally not seen with non-dividend paying stocks. The reason is that you could then simply go short in the spot market, go long in futures, and at maturity you get the stock for the agreed-upon futures price. You then clear the short position, and gain the differential. The reason why this does not normally create arbitrage opportunities in the case of commodities is the convenience yield.
The opposite relationship is, as mentioned earlier, called contango, where the futures price is higher than the current spot price.
Keynes (1930) explained backwardation through a risk premium. The spot price should exceed the futures or forward price by a certain risk premium that the hedger is willing to pay in order to minimize price risk. In the equation above, the risk pre- mium is directly incorporated through the risk-neutral probability measure, which means that the risk premium (e.g. the difference between the expected spot price and the futures price) is zero (Eydeland and Wolyniec, 2003). When talking about
commodity or financial pricing, we can consider this risk-neutral world instead of the physical world, where no arbitrage and a risk premium of zero is assumed. Even though this is true on average under the risk neutral measure, the spread, or the risk premium, might be large such that the holder of the futures contract might face substantial basis risk (Eydeland and Wolyniec, 2003).
3.2 Risk Management of Commodities
In today’s volatile markets, companies need to hedge their positions, whether they are committed to buying or selling a commodity. Hedging, more specifically, means taking a position that offsets the risk faced by an initial, risk-exposed position. Risk management tools provide more certainty about future revenues and expenses for larger firms that want to control their cash flows, meet their operational needs and ensure their funding for future projects. Given this volatile nature of the markets, risk management plans should be reviewed often, so that the hedge is always optimal and matches the current market situation.
3.2.1 Exchange-Traded Products vs. OTC
When talking about risk management tools, one separates between standardized fu- tures markets and off-exchange over-the-counter (OTC) markets. The OTC market offers more customized products such as forwards, swaps and other options. Because futures markets are rather liquid, this development supported the rise of many dif- ferent OTC instruments, where swaps, options on futures and other derivatives are a few examples (Fusaro, 1998). This provided even better and more customized risk tools for firms to use when hedging against adverse price movements.
While futures contracts are rather short-termed, OTC contracts can be agreed upon for both short and long maturities, ranging from a couple of months to several years forward. One concern is credit risk. With exchange-traded products, one is nearly certain that the counterparty will meet his or her obligation due to the margins paid, e.g. losses on futures positions are settled daily. One does not have this safety when trading off-exchange. However, clearinghouses have in later years become part of the OTC market, where their task is to ensure that the contracted parties obey their payments. The risk is also higher with very long-termed OTC contracts, as it is nearly impossible to predict what will happen far into the future.
Initially, crude oil dominated the OTC market, but refined products and gas soon caught up (Fusaro, 1998). OTC markets will probably grow in importance and become more liquid, and futures markets will complement this development by providing transparent price quotes that can be used as benchmarks for OTC deals.
3.2.2 Different Types of Risk
Price risk refers to the risk coming from adverse movements in prices (Fusaro, 1998).
As a producer of a commodity, for example, you have a commitment to sell the commodity in the spot market. The price risk for the producer lies in the possible decrease in spot prices. As a consumer, you need to buy the commodity. In his or her case, the risk lies in increasing spot prices.
Credit risk refers to the risk that the counterparty in any transaction is not able to fulfill his or her obligations stated in the contract, i.e. not being creditworthy.
When talking about forward and futures contracts, there might be a larger credit risk in a forward contract than that of a futures contract. The reason is simple: if the forward contract becomes more in the money (the difference between the spot price and the forward price increases) throughout the period, then the party being long (the holder of the contract) faces higher counterparty credit risk (Eydeland and Wolyniec, 2003).
Even though a futures contract might partly or wholly eliminate price and credit risk, it does not mitigate the so-calledbasis risk (Fusaro, 1998). It is assumed that the futures price will move towards the underlying spot price of the same commodity as it approaches maturity. This is however not always the case. Basis risk becomes highly relevant when talking about cross or proxy hedging. In Denmark, for example, there is no liquid forward market for natural gas. In order for a Danish energy company to hedge its commitment to sell gas in the Danish spot market, they must go abroad in order to hedge with forwards or futures. It is then of the essence that the foreign market is correlated to the Danish natural gas market - only then will the hedge be effective given regional differences in natural gas markets. The degree of correlation between different regions depends on the local demand, local production, local availability with respect to pipelines etc. (Fusaro, 1998). The basis is defined as
S(t)−FT(t), (3.2.1)
where S(t) is the spot price at time t, and FT(t) is the futures price at time t for a contract maturing at time T, with t < T. The basis risk is often defined as the variance of the basis, and in general it is random. A random basis means that the basis risk cannot be completely eliminated (Duffie, 1989). The basis risk is zero when the variances of the spot price and the futures price are equal, and their correlation is 1 (e.g. they are perfectly correlated). The variance of the basis is
V ar[S(t)−FT(t)] = V ar[S(t)] +V ar[FT(t)]−2Cov(S(t), FT(t)).
Now, given thatρ= Cov(S(t),Fσ T(t))
S(t)σF T(t) , we get the variance of the basis to be V ar[S(t)−FT(t)] = V ar[S(t)] +V ar[FT(t)]−2ρσS(t)σFT(t).
Given thatV ar[S(t)] =V ar[FT(t)], which impliesσS(t)=σFT(t), andρ= 1, we can rewrite as
V ar[S(t)−FT(t)] = 2V ar[S(t)]−2σ2S(t)= 0.
So the variance of the basis is zero when the hedged commodity is perfectly correlated to the futures price. Even with non-zero basis risk, it is still interesting to see how effective the hedge is. An example of how to measure the hedge effectiveness is
h= 1− V ar[S(t)−FT(t)]
V ar[S(t)] . (3.2.2)
The closer h is to 1, the better is the hedge with respect to eliminating price risk.
The Danish energy company that has a commitment to sell gas in the Danish spot market can hedge against a drop in the prices by going short in German natural gas futures through the Intercontinental Exchange (ICE). Say that the producer needs to hedge a spot commitment in three months, and therefore shorts (or sells) ICE futures maturing in three months. But since gas prices in Denmark probably are not perfectly correlated to the German ones, the futures price will not exactly converge to the Danish spot price at maturity. This is the basis risk, the risk that the price of the commodity being hedged does not fluctuate in the exact same manner as the commodity one hedges with.
In order to hedge this basis risk, the Danish energy company could make an OTC swap agreement, where they sell the basis in a swap three months forward. If Danish prices then dropped faster than the ICE prices, the risk is covered. Another issue here is also the different currencies, which could have been dealt with by an OTC
broker. As always, one has to consider the counterparty credit risk when entering into OTC agreements, and also the more likely high premiums that have to be paid in order to hedge this position completely.
Worth mentioning in this subsection is the Exchange for Physical (EFP). There are several ways of ending a futures contract; entering an offsetting futures position, physical delivery at expiration or EFP (Eydeland and Wolyniec, 2003). An EFP integrates the OTC basis market, the physical market and the futures market. From our example, the Danish energy company might have a commitment to sell gas to a German energy company at a premium over the ICE futures, where the price is fixed by choice of the German company. To hedge against a decrease in prices, the Danish company sells (short) ICE futures. The German company buys (long) ICE futures. When the delivery takes place, the two companies switch futures positions.
The German company sells futures to the Danish company at the agreed price (or premium over ICE), liquidating the futures positions on both sides and receives the physical commodity. The German company’s long position is exchanged for the physical supply (ICE, 2008).
3.2.3 Hedging with Futures
Forwards and futures contracts are probably the most common way of managing risk, and much research has been done within this field.
An energy company who has a commitment to sell a commodity in the spot market at a future date is exposed to price risk and does not want prices to drop.
It could therefore lock in a minimum profit by entering into a short futures commit- ment, e.g. selling futures. That way the company will receive the contracted futures price for the lot. The downside to this agreement is that the energy company will forego potential profits if the spot price were to increase. At the opposite side, the end consumer might worry about an increase in prices. He or she could therefore secure his or her expenses by entering into a long futures commitment, e.g. buy- ing futures. That way a ceiling is placed on future expenses. When hedging with futures, there are three main decisions we must make:
1. Short or long position 2. Size of the position 3. Timing of the position
The first decision has been well covered and is basically a question of whether one has the commitment to sell or buy in the spot market. Ideally, if one has a commitment in the spot market in three months, one would take an opposite position in a futures contract maturing in three months. If these two dates do not perfectly match, we are left with some basis risk. In perfect and arbitrage-free markets, the spot price should coincide with the futures price at maturity. In many cases, however, there is a mismatch between the two dates and thus between the spot and futures price at delivery (Duffie, 1989). The basis risk normally increases with the difference between the two dates, and the best hedge is the one where the two dates are as close as possible.
One option when the commitment date and the futures’ maturity date do not coincide is to choose a futures contract maturing a bit earlier than the commitment date and then roll over the hedge by entering into a new futures contract maturing later. By choosing a futures contract that matures at a later date than the com- mitment date itself, one eliminates the need to roll over. This might be beneficial in some cases, but in others it might be better to choose the roll-over option. As already mentioned, liquidity seems to decrease with time to maturity. There might thus be some liquidity costs involved in choosing the contracts maturing at a later date, since those contracts often are less liquid. Liquidity costs would in this case be due to the scarcity of accommodating traders. With a lower trading volume (e.g.
lower liquidity), the greater is the risk for the accommodating trader, and he would thus require a larger premium in order to take the opposite futures position. As Duffie (1989) explains, there might be higher premiums (e.g. a high bid-ask spread for a market order to buy) during periods of low trading activity. The higher bid-ask spread in the contracts maturing later might lead hedgers to prefer the contracts maturing sooner.
If the two dates coincide the position that would eliminate risk the best is an equal and opposite position in the futures market. But given the regular mismatch between the two, it is hard to eliminate all risk. Optimally, the hedge ratio would look like this
h = −Size of Spot Commitment ×β, (3.2.3) where
β = Covariance(∆S,∆F)
Variance(∆F) = Corr(∆F,∆S)× SD(∆S)
SD(∆F), (3.2.4)
and where ∆S = St+1 −St and ∆F = Ft+1 −Ft. If the spot commitment date equals the maturity date for the futures contract, the basis is zero (Duffie, 1989).
This would give us a β equal to one, such that h equals the opposite size of the spot commitment. As already mentioned, when the basis is random, one cannot eliminate all risk as one could have done in the case of a basis equal to zero.
The risk is smallest when choosing the futures position h, which minimizes the variance of the value of the position. β is the hedge coefficient. When the correlation between the change in futures and spot prices is large, then the optimal futures hedge is larger. Logically, the higher the correlation between spot and futures prices, the more one can gain or loose, and thus the need for larger hedge. β is also increasing with the standard deviation of the spot price change, meaning that the more volatile spot prices, the larger the necessary hedge. The same logic applies in the latter example.
Everything mentioned above is seen from a static point of view, with discrete time points. A more sophisticated and optimal hedge would be adjusted dynamically. As each day or point in time passes, new information is available and the hedge should be adjusted accordingly. This would tackle the time-changing risk exposures of many firms and individuals due to the time-changing volatilities of many commodities (Duffie, 1989). Calculating the optimal dynamic hedge might however be quite challenging, but will be dealt with in Chapter 6.
Other than forwards and futures contracts, there exists a plethora of derivatives that can be used for hedging purposes. Among them are swaps, options on futures, spreads and spread options, caps and floors etc. These will not be explored in further detail in this thesis, since they are not used in the hedging analysis later on.
3.3 Optimal Futures Hedge Ratios
When futures contracts are used for hedging purposes, it is, as stated earlier, nec- essary to decide on how many futures contracts to purchase (sell). The hedge ratio gives the number of futures contracts needed to hedge the spot position. If it is big- ger than 1, this means that the futures position is larger than the spot position, and vice versa. How the optimal hedge ratio (OHR) is calculated depends on the under- lying objective function being maximized or minimized (Chen, Lee, and Shrestha, 2003). A typical na¨ıve hedge is to take an exact opposite position in futures, such
that the hedge ratio is 1. However, changes in the prices of futures do often not perfectly match those of the spot, and a na¨ıve hedge does not eliminate all the risk lying in the spot position. At best, and in most cases, it reduces the risk compared to an unhedged position (Junkus and Lee, 1985).
Probably the most common objective function is the variance of the cash flows of the hedged portfolio, which is to be minimized (Junkus and Lee, 1985; Ederington, 1979; Johnson, 1960). It is rather easy to compute and interpret, but instead of incorporating the expected return of the hedged portfolio, it only minimizes its variance. As we will see, this hedging solution is only consistent with the mean- variance hedge if individuals are infinitely risk averse, or if the futures price process is a martingale.
Objective functions that take both expected return and risk into account are the so-called mean-variance OHRs (Howard and D’Antonio, 1984). If the futures price process is a martingale in this case, the optimal mean-variance hedge ratio will be equal to the minimum variance ratio. For the OHR to be consistent with the objective of maximizing the expected utility, either the underlying utility has to be of the quadratic type, or returns must be jointly normal (Chen, Lee, and Shrestha, 2003).
What makes this complicated, is the fact that one needs to assume a certain utility function and return distribution. Several researchers have tried to ease these assumptions, by for example minimizing the mean extended-Gini (MEG) coeffi- cient, or by minimizing the generalized semi variance (GSV). Both of these methods are consistent with the so-called stochastic dominance concept (Chen, Lee, and Shrestha, 2003). The idea is that one situation can be ranked above another, based on preferences with respect to possible outcomes.
What is also important to consider, is whether to estimate a static (fixed) OHR or one of the time-varying kind. A static hedge is undertaken at the beginning of the hedging horizon, and kept fixed until the period is over. A dynamic hedge is initiated at the beginning of the hedging horizon, but instead of being kept fixed it is revised and updated periodically during the hedging horizon. As new information reaches the hedger, the hedging position can be updated accordingly. Whether the updating is done continuously or discrete depends on the hedgers motive. A semi- dynamic strategy would be to consider a multi-period model, where the hedge ratio is updated at discrete time points. As these hedge ratios, both semi-dynamic and
dynamic, are updated conditional on new information, the moments underlying the calculation are conditional. For a static hedge ratio calculation, the moments are unconditional.
The most common way of estimating the static hedge ratio is based on the or- dinary least squares (OLS) regression (Ederington, 1979). When estimating the dynamic version, conditional models such as the autoregressive conditional het- eroscedastic (ARCH), the generalized ARCH (GARCH) or the cointegration meth- ods can be used (Chen, Lee, and Shrestha, 2003).
For the following subsections, we will consider a general setup concerning the cash flows of a particular firm. This setup is in accordance with the article of Chen, Lee, and Shrestha (2003), and also many other articles describing the same topic.
A firm wants to hedge a commitment in the spot market to either buy or sell some commodity (or financial asset). The commitment is inNs units of the spot product, and the hedge consists of Nf units of futures. St and Ft are the spot and futures prices respectively at time t. The Nf units of futures contracts are entered into so that fluctuations in the spot position will be reduced. Say the portfolio consists consists of Ns units of a long spot position, and of Nf units of a short futures position. This is the hedged portfolio, and its profit ∆VH and the hedge ratioH are
∆VH =Ns∆St−Nf∆Ft and H = Nf
Ns, (3.3.1)
where ∆St=St+1−St and ∆Ft=Ft+1−Ft.
Instead of expressing the hedged portfolio in terms of its profit, we can formulate the hedged portfolio’s return as
Rh = NsStRs−NfFtRf
NsSt =Rs−hRf, (3.3.2) where h = NNfFt
sSt is the hedge ratio, and Rs = St+1S−St
t and Rf = Ft+1F−Ft
t are the one-period returns on the spot and futures positions, respectively.
The hedger’s main concern is to choose H or h that minimizes risk, or fluctuations in the spot position, and below we will discuss the different strategies based on what becomes relevant later in this thesis.
3.3.1 Minimum Variance Hedge Ratio
The minimum variance (MV) hedge ratio was derived by Johnson (1960) by minimiz- ing the variance of the hedged portfolio’s profit. The profit of the hedged portfolio
and the hedge ratio are, as in (3.3.1)
∆VH =Ns∆St−Nf∆Ft and H = Nf Ns. The variance of ∆VH is
V ar(∆VH) =Ns2V ar(∆St) +Nf2V ar(∆Ft)−2NsNfCov(∆S,∆F).
By minimizing this variance with respect to H, the MV hedge ratio becomes H∗ = Nf
Ns = Cov(∆S,∆F)
V ar(∆F) . (3.3.3)
If we choose to work with the return of the hedged portfolio instead, the variance of Rh is
V ar(Rh) =V ar(Rs) +h2V ar(Rf)−2hCov(Rs, Rf),
and the corresponding hedge ratio, calculated in the same way as above, becomes h∗1 = Cov(Rs, Rf)
V ar(Rf) =ρσs
σf, (3.3.4)
where ρ is the correlation between Rs and Rf, and σs and σf are the standard deviations ofRs and Rf.
Ederington (1979) estimates the MV hedge ratio H∗, and looks at the hedging effectiveness as follows
E = 1− V ar(Rh)
V ar(Rs). (3.3.5)
The closer E is to 1, the more effective is the implemented hedge.
The most common way of estimating this static MV hedge ratio is by running a simple ordinary least squares (OLS) regression:
∆St=c+H∆Ft+t. (3.3.6)
The changes in spot prices are regressed on changes in futures prices. If choosing to work with returns instead of cash flows, the OLS regression becomes:
∆Rs =c+hRf +t. (3.3.7)
As is widely known and sited in many econometrics textbooks, for an OLS es- timate to be unbiased and optimal, several conditions need to be satisfied (e.g.
