Proxy Hedging of Commodities

Transcription

1 Proxy Hedging of Commodities Hedging Gasoil Commitments in Related Futures Markets Master s Thesis in Advanced Economics and Finance 1 by Cecilie Viken and Marianne Solem Thorsrud November 16, 2013 Supervisors Nina Lange 2 and Kristian R. Miltersen 3 Department of Finance Copenhagen Business School 1 Number of Characters (Pages): 269,133 (118) 2 nl.fi@cbs.dk 3 krm.fi@cbs.dk

2 Copyright c November 2013 Cecilie Viken and Marianne Solem Thorsrud This thesis was written using the typewriting program L A TEX, 12pt.

3 Executive Summary 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 utility hedges and a naïve 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 semidynamic 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. i

4 Aknowledgements 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 markets 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 Viken 4 Marianne Solem Thorsrud 5 Oslo, November viken cecilie@hotmail.com 5 marianne.thorsrud@gmail.com ii

5 Contents Executive Summary Aknowledgements i ii 1 Introduction Introducing the Problem in Question Organization of the Thesis Overview of Energy Markets Introduction to Energy Markets Characteristics of Energy Prices The Crude Oil Market The Value of Crude Oil Trading North Sea Oil The Gasoil Market Refining Crude Oil The European Gasoil Market Review of Futures Markets and Hedging Futures Markets Forward and Futures Contracts Equilibrium in the Futures Market Liquidity Futures Prices Risk Management of Commodities Exchange-Traded Products vs. OTC Different Types of Risk iii

6 CONTENTS iv Hedging with Futures Optimal Futures Hedge Ratios Minimum Variance Hedge Ratio Utility-Based Hedge Ratios Dynamic Hedge Ratios Proxy Hedging Liquidity versus Correlation Long-Term Relationship Short Summary Modeling of Prices Introduction Stochastic Processes Wiener Process Generalized Wiener Process Itô Processes and Itô s Lemma Spot Price Models Geometric Brownian Motion The 1-factor Model Gibson-Schwartz 2-factor Model Relationship between Spot and Futures Prices Risk Neutral Probabilities Pricing of Futures Contracts The 3-factor Spot Price Model Rationale Data The 3-Factor Model General Setup Cholesky Decomposition of Correlations System of Equations to be Simulated State-Space Models and the Kalman Filter Estimation and Results Simulation of Prices Discussion

7 v CONTENTS 6 Proxy Hedging General Setup Minimum Variance Hedge Mean-Variance Hedge Naïve Hedge Effectiveness Hedging Strategies Static Hedge Static Rolling Hedge Semi-Dynamic Hedge Discussion Case: DONG Energy buying Gas Continental European Term Contracts Contract Setup Results Conclusion 120 Appendix A Figures, Tables and Mathematical Proofs 122 A.1 Figures A.2 Tables A.3 Mathematical Proofs Appendix B R Codes 137 B.1 Data Description B.2 Simulations of Wiener Processes B.3 Stationarity Tests of Data B.4 Kalman Filter - FKF Package B.5 GARCH(1,1)-M Modeling Bibliography 158

8 Chapter 1 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 Miltersen 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 companies. 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 hedging 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. 1

9 1.1. Introducing the Problem in Question 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 deregulated 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 introduction to energy markets, with a focus on the European crude oil and gasoil markets.

10 3 1. Introduction 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 Chapter 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 hedging 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.

11 Chapter 2 Overview of Energy Markets In this chapter we give an overview of energy markets. Although the term 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 information 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 4

12 5 2. Overview of Energy Markets 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, forwards, futures, swaps and options. Forwards is referred to as bilateral agreements and are traded over-the-counter (OTC), whereas futures are standardized agreements and are concluded at a commodity exchange. The most significant exchanges from a global perspective is the New 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 International 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 The first contract offered by the IPE was gasoil futures, but the Brent crude futures contract followed in Today, the ICE offers, amongst other contracts, the benchmarks Brent crude oil, West Texas Intermediate (WTI) crude oil and gasoil futures contracts (the Intercontinental Exchange) Crude Oil.pdf

13 2.2. Characteristics of Energy Prices 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 the seasonal 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 sudden spikes or jumps 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 called mean reversion of prices. This concept of whether to store the commodity or buy extra from elsewhere

14 7 2. Overview of Energy Markets 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 current term structure of the prices, that is, whether the market is in contango or backwardation. The market will experience contango when forward prices are higher than spot price, and backwardation when they are lower than spot prices (Schofield, 2007). For instance crude oil is more prone to backwardation. The reason is that

15 2.3. The Crude Oil Market 8 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 commodity, 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 commodity, 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 worldwide energy consumption and 30% of European consumption. Although the majority 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 2.1 2, 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. 2 Figure taken from Burger, Graeber, and Schindlmayr (2008) and Crude Oil.pdf

16 9 2. Overview of Energy Markets Figure 2.1: Historical Brent Spot Prices 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 between heavy 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

17 2.3. The Crude Oil Market 10 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 a floating price, prices are set relative to futures prices and two counterparties agree to do a physical trade of oil and define the price as an exchange 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 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 and Ekofisk (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

18 11 2. Overview of Energy Markets 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 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

19 2.4. The Gasoil Market 12 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 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 are gasoline, 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 the gross 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.

20 13 2. Overview of Energy Markets The European Gasoil Market In Europe, the refining industry is defined by geography and is divided into ARA (Amsterdam-Rotterdam-Antwerp) and Mediterranean (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 demand. However, due to refining capacities, supply does not always meet demand 5. Some refineries have excess capacity, producing more than local demand, and others lower capacity thus not meeting demand. costs or high discounts and premiums when trading. This may result in higher transaction The reason for this asymmetry between refining capacity and consumer demand is that expanding capacity takes much more than just increasing production; the whole infrastructure, including 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 introduced 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 Russia, 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 Gasoil Brochure.pdf 5 Traders Should Buy and Sell Refinery Gasoil 6 Gasoil Brochure.pdf

21 2.4. The Gasoil Market 14 below (Intercontinental Exchange) 7 : Figure 2.2: Example of Gasoil Futures Contract Characteristics 7 Gasoil Brochure.pdf

22 Chapter 3 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 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

23 3.1. Futures Markets 16 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 contracts, simply because we later on will assume a constant interest rate 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

24 17 3. Review of Futures Markets and Hedging 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 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.

25 3.1. Futures Markets 18 (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.

26 19 3. Review of Futures Markets and Hedging (a) Open Interest ARA gasoil Futures (b) Volume ARA gasoil Futures Figure 3.2: Liquidity of ARA gasoil Futures 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 maturity increases, or equivalently; more volatile as expiration approaches. This is the so-called Samuelson Effect, which Paul A. Samuelson discovered in This argu-

27 3.1. Futures Markets 20 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 available 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 expiration 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 time t, of a commodity that will be sold / received at time T, equals today s expected spot price under Q 1 at time T, then we can predict future spot prices through the futures price. F t (T ) = E Q (r δ)(t t) t [S T ] = S t e when t T 1 More information about the risk-neutral Q-measure will be given in Chapter 4, Section 4.4

28 21 3. Review of Futures Markets and Hedging 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 S T underlying the futures contract. In the real world, this expected value of S T 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 autocorrelation, 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 F i = F 1, F 2,..., F n 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 premium 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

29 3.2. Risk Management of Commodities 22 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 Exchange-Traded Products vs. OTC When talking about risk management tools, one separates between standardized futures 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 different 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.

30 23 3. Review of Futures Markets and Hedging 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 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-called basis 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) F T (t), (3.2.1)

31 3.2. Risk Management of Commodities 24 where S(t) is the spot price at time t, and F T (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) F T (t)] = V ar[s(t)] + V ar[f T (t)] 2Cov(S(t), F T (t)). Now, given that ρ = Cov(S(t),F T (t)) σ S(t), we get the variance of the basis to be σ F T (t) V ar[s(t) F T (t)] = V ar[s(t)] + V ar[f T (t)] 2ρσ S(t) σ F T (t). Given that V ar[s(t)] = V ar[f T (t)], which implies σ S(t) = σ F T (t), and ρ = 1, we can rewrite as V ar[s(t) F T (t)] = 2V ar[s(t)] 2σ 2 S(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) F T (t)]. (3.2.2) V ar[s(t)] 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

32 25 3. Review of Futures Markets and Hedging 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) 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 commitment, 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. buying 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

33 3.2. Risk Management of Commodities 26 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 commitment 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)

34 27 3. Review of Futures Markets and Hedging and where S = S t+1 S t and F = F t+1 F t. 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, necessary 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 bigger 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 underlying objective function being maximized or minimized (Chen, Lee, and Shrestha, 2003). A typical naïve hedge is to take an exact opposite position in futures, such

35 3.3. Optimal Futures Hedge Ratios 28 that the hedge ratio is 1. However, changes in the prices of futures do often not perfectly match those of the spot, and a naïve 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 meanvariance 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) coefficient, 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 semidynamic 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

36 29 3. Review of Futures Markets and Hedging 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 ordinary least squares (OLS) regression (Ederington, 1979). When estimating the dynamic version, conditional models such as the autoregressive conditional heteroscedastic (ARCH), the generalized ARCH (GARCH) or the cointegration methods 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 in N s units of the spot product, and the hedge consists of N f units of futures. S t and F t are the spot and futures prices respectively at time t. The N f units of futures contracts are entered into so that fluctuations in the spot position will be reduced. Say the portfolio consists consists of N s units of a long spot position, and of N f units of a short futures position. This is the hedged portfolio, and its profit V H and the hedge ratio H are V H = N s S t N f F t where S t = S t+1 S t and F t = F t+1 F t. and H = N f N s, (3.3.1) Instead of expressing the hedged portfolio in terms of its profit, we can formulate the hedged portfolio s return as R h = N ss t R s N f F t R f N s S t = R s hr f, (3.3.2) where h = N f F t N ss t is the hedge ratio, and R s = S t+1 S t S t and R f = F t+1 F t F t one-period returns on the spot and futures positions, respectively. are the 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 Minimum Variance Hedge Ratio The minimum variance (MV) hedge ratio was derived by Johnson (1960) by minimizing the variance of the hedged portfolio s profit. The profit of the hedged portfolio

37 3.3. Optimal Futures Hedge Ratios 30 and the hedge ratio are, as in (3.3.1) V H = N s S t N f F t and H = N f N s. The variance of V H is V ar( V H ) = N 2 s V ar( S t ) + N 2 f V ar( F t ) 2N s N f Cov( S, F ). By minimizing this variance with respect to H, the MV hedge ratio becomes H = N f N s = Cov( S, F ). (3.3.3) V ar( F ) If we choose to work with the return of the hedged portfolio instead, the variance of R h is V ar(r h ) = V ar(r s ) + h 2 V ar(r f ) 2hCov(R s, R f ), and the corresponding hedge ratio, calculated in the same way as above, becomes h 1 = Cov(R s, R f ) V ar(r f ) = ρ σ s σ f, (3.3.4) where ρ is the correlation between R s and R f, and σ s and σ f deviations of R s and R f. are the standard Ederington (1979) estimates the MV hedge ratio H, and looks at the hedging effectiveness as follows E = 1 V ar(r h) V ar(r s ). (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: S t = c + H F t + ɛ 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: R s = c + hr f + ɛ t. (3.3.7) As is widely known and sited in many econometrics textbooks, for an OLS estimate to be unbiased and optimal, several conditions need to be satisfied (e.g.

38 31 3. Review of Futures Markets and Hedging Wooldridge, 2010). The OLS estimator is consistent when the regressors are exogenous and when perfect multicollinearity is non-existent. It is BLUE (Best Linear Unbiased Estimator) when errors are homoscedastic and not autocorrelated. If errors are heteroscedastic, this violates the assumption of homoscedasticity, and the OLS estimator is not the best estimator. ARCH and GARCH models might solve this problem by taking into account heteroscedastic errors. By using these conditional methods, the hedge ratio can also be updated over time giving it a dynamic nature. Baillie and Myers (1991) and Cecchetti, Cumby, and Figlewski (1988) consider a bivariate GARCH model to estimate their optimal futures hedge ratios. The model is bivariate since two markets are considered; the spot and futures markets. The model looks like this: [ ] St = F t [ µ1 µ 2 ] [ ] [ ] ɛ1t H11,t H 12,t + where ɛ t Ω t 1 N(0, H t ), H t = ɛ 2t H 12,t H 22,t (3.3.8) and vec(h t ) = C + A vec(ɛ t 1 ɛ t 1) + B vec(h t 1 ). In this case, the conditional MV hedge ratio is calculated as h t 1 = H 12,t /H 22,t. Since H t in this example basically is the conditional variance-covariance matrix, h t 1 can equivalently be calculated as h t 1 = Cov( S, F ) Ω t 1 V ar( F ) Ω t 1. Instead of considering a bivariate GARCH, the model can be multivariate with several spot and futures contracts. Another relevant issue when dealing with time series is stationarity. According to Tsay (2010), a time series {S t } is weakly stationary if the mean of S t and the covariance between S t and S t l are both time-invariant. l is an arbitrary integer. A plot of the time series would typically then show us a series that fluctuate with constant variation around a constant level. If for example two time series are nonstationary, a linear combination of the two can still be stationary, which relates to the concept of cointegration. Often this is referred to as a long-term relationship between the two cointegrated time series (Engle and Granger, 1987). If the two time series considered are cointegrated, then the OLS regression from equation (3.11) will be mis-specified. A cointegration analysis and an error-correction model must then be considered. First, one has to test for unit roots in the time series considered.

39 3.3. Optimal Futures Hedge Ratios 32 This can easily be done by for example running the Augmented Dickey-Fuller - test (Dickey and Fuller, 1981) using programming software such as R or Stata. If both series contain a unit root, then a cointegration test is performed (Engle and Granger, 1987). If the series are cointegrated, a new regression other than the OLS has to be specified. Following the structure of Chen, Lee, and Shrestha (2003), we define a regression of the following cointegrated form: S t = c + hf t + u t. (3.3.9) First, the regression above is estimated. Second, we estimate an error correction model (ECM), where we explicitly incorporate the errors from the regression: S t = ρu t 1 + β F t + m n δ i F t i + θ i S t j + ɛ j. (3.3.10) i=1 j=1 Having estimated the latter ECM, we obtain an estimate of the hedge ratio, β. Several ECMs can be used, and Lien and Luo (1993) do for example specify (S t F t ) as being the long-term relationship. u t 1 is then replaced by (S t 1 F t 1 ) Utility-Based Hedge Ratios The so-called mean-variance hedge ratio is calculated based on both risk and return, and assumes either a quadratic utility function, or jointly normally distributed returns. As Chen, Lee, and Shrestha (2003) write, Hsin, Kuo, and Lee (1994) maximize the utility function below in order to get an optimal hedge ratio max N f U(E(R h ), σ; λ) = E(R h ) 0.5λσ 2 h, (3.3.11) where λ is the risk aversion parameter. This strategy is consistent with the meanvariance objective, but we need some knowledge about the risk aversion parameter in order to derive the optimal hedge ratio. Individuals or firms that are highly risk averse will have a high λ, and vice versa. The hedge ratio in this case becomes: [ ] h 2 = N ff N s S = E(R f ) ρ σ s. (3.3.12) λσf 2 σ f As can be seen, if E(R f ) = 0, meaning that the futures price process is a martingale such that expected return is zero, or if λ, meaning that the individual or firm

40 33 3. Review of Futures Markets and Hedging is infinitely risk-averse, then h 2 will equal h 1, which is the minimum variance hedge ratio. The hedge ratio above can be estimated by using the moments from the data sample we possess, such that we use the average returns, sample standard deviation and sample correlation. In the case above, a quadratic utility function or returns that are jointly normally distributed was assumed. Howard and D Antonio (1984), following a closely related path, derived an OHR and an effectiveness measure based on the Sharpe index. The hedger in their paper can form a portfolio by investing in the spot asset, futures contracts and a risk-free asset. The hedger s one-period maximization problem is to maximize θ = E(Rp) i σ p, where E(R p ) is the expected return on the portfolio, i is the risk-free rate of return and σ p is the standard deviation of the return on the portfolio. More specifically, E(R p ) = N sp s E(r s ) + N f P f E(r f ) and N s P s σ p = 1 Ns N s P 2 Ps 2 σs 2 + Nf 2P f 2σ2 f + 2N sn f P s P f σ s σ f ρ, s where the notations above are similar to the ones mentioned earlier, and E(r s ), E(r f ) are expected one-period returns of the spot and futures prices. developed as max N f E(R p ) i σ p. The problem is By maximizing this expression wrt. N f, after having replaced for E(R p ), they find that N f = N sb, where b is the OHR: b E(r f )σ s /[E(r s ) i]σ f ρ = P f σ f /P s σ s (1 (E(r f )σ s /[E(r s ) i]σ f )ρ) λ ρ = γπ(1 λρ), (3.3.13) and where λ = E(r f )σ s [E(r s) i]σ f, π = σ f σ s and γ = P f P s. Howard and D Antonio (1984) further define the hedging effectiveness as one that incorporates the maximum slope of the capital market line in the more general capital asset pricing model (CAPM). They derive a closed-form solution of the hedging effectiveness that depends on both risk and return, which is also the main contribution of this paper: 1 2λρ + λ HE = 2. (3.3.14) 1 ρ 2

41 3.3. Optimal Futures Hedge Ratios 34 In order to investigate other utility-based hedge ratios, we need to assume other underlying utility functions. Cotter and Hanly (2012) look at several different utility functions, and derive the corresponding hedge ratios. As mentioned above, it is necessary to make certain assumptions regarding the risk aversion of the hedger. Absolute risk aversion (ARA) is an absolute measure of risk aversion, meaning that it is a measure of how the hedger reacts to dollar changes in wealth W. The coefficient of absolute risk aversion is defined as ARA = U (W ) U (W ). (3.3.15) Relative risk aversion (RRA) is a measure of how the hedger reacts to relative changes in W, and is defined as RRA = W U (W ) U (W ). (3.3.16) If the ARA is increasing in wealth W, then the amount of risky assets held in the portfolio will be reduced if W goes down. If RRA is increasing in W, then the amount of risky assets held in the portfolio relative to the risk-free assets (i.e. the proportion of risky assets) will be reduced if W goes down. decreasing in wealth, the opposite generally holds. If ARA or RRA is According to Cotter and Hanly (2012), who continue their analysis focusing on the RRA, the quadratic utility function is defined as U(W ) = W aw 2, (3.3.17) where a is a positive measure of risk aversion. The first derivative with respect to W, U (W ) = 1 2aW is defined to be > 0, since in general more is preferred to less the individual or firm. The second derivative is U (W ) = 2a. The measure of relative risk aversion then becomes R R (W ) = 2aW, and that of absolute risk 1 2aW aversion is R A (W ) = 2a. As W increases, R 1 2aW A(W ) increases, so that the amount invested in risky assets decreases. The proportion invested in risky assets relative to risk-free assets decreases since R R (W ) increases in W. The OHR is shown to be equal to h 3 = E(r f,t) 2λσ 2 f,t + σ sf,t, (3.3.18) σf,t 2 where E(r f,t ) is expected return on the futures position, λ is the risk aversion parameter, σ 2 f,t is the variance of the futures and σ sf,t is the covariance between the

42 35 3. Review of Futures Markets and Hedging spot and futures. As the risk aversion parameter λ increases (decreases), the first term in the OHR becomes larger (smaller). As can be seen, if E(r f,t ) = 0, meaning that the futures price process is a martingale such that the expected return is zero, or if λ, meaning that the individual or firm is infinitely risk averse, then h 3 will equal h 1, which is the minimum variance hedge ratio. h 3 found in this case is almost exactly the same OHR as h 2 found above, in equation (3.3.14), except here we divide the speculative term by 2. Conlon, Cotter, and Gencay (2012) incorporate a range of investor preferences into their horizon dependent mean-variance hedging strategy. They underline the importance of including risk aversion preferences and firm-specific expectations regarding future returns into the hedging analysis. They call it speculative hedging. The classical MV strategy assumes that future prices are martingales, i.e. that the best predictor of future prices is today s price, implying an expected return of zero. Their paper deviates from this belief in that it assumes non-zero expected returns. The formulas to follow are as derived in the paper by Conlon, Cotter, and Gencay (2012). By looking at several different values for λ, by examining different hedging horizons 2 and by imposing certain assumptions regarding the expected returns, they compare the OHRs. They find that for positive expected returns, the OHR decreases for a low λ, and in this case the OHR is smaller when comparing long to short hedging horizons. For negative expected returns, the OHR increases, still for a low λ. Hedging effectiveness is largest when positive expected returns were correctly forecasted over a long hedging horizon. For large values of λ, e.g. high risk aversion, the OHR converges to the MV OHR the expected return. Hedging effectiveness is largest for long hedging horizons. The log utility function is defined as U(W ) = ln(w ), (3.3.19) where the first derivative is U (W ) = 1/W and the second derivative is U (W ) = 1/W 2. The measure of relative risk aversion is R R (W ) = W 1/W 2 R A (W ) = 1/W. 1/W = 1, and Since the relative risk aversion is constant and equal to 1, the proportion invested in risky assets relative to risk-free will stay constant W. The OHR in the log utility-case, with λ = 1, is shown to be equal to h 4 = E(r f,t) 2σ 2 f,t + σ sf,t. (3.3.20) σf,t 2 2 This is done through a wavelet transformation ; see the article for method and explanations

43 3.3. Optimal Futures Hedge Ratios 36 The exponential utility function is defined as U(W ) = e aw, (3.3.21) where a > 0, the first derivative is U (W ) = ae aw and the second derivative is U (W ) = a 2 e aw. The measure of relative risk aversion is R R (W ) = W a and that of absolute risk aversion is R A (W ) = a. The relative risk aversion is increasing in W, and the absolute risk aversion is constant and equal to a. The OHR will not be covered in this case, since it is not used further on in the analysis. Common to all OHRs above based on utility functions is the need of an estimate of λ, as well as sample moments of the data we have. What Cotter and Hanly (2012) do is that they estimate the coefficient of RRA based on a market risk premium for oil and gas producers in the energy industry. This premium is defined as the excess return on a portfolio of assets that is required to compensate for systematic risk : E(r p,t ) rf = λσp,t, 2 (3.3.22) where the left-hand side is the risk premium on the market portfolio, λ is the coefficient of RRA and σp,t 2 is the variance of the return in the market. They further define the equation above as r p,t ɛ t = E(r p,t ) rf, where r p,t is the return on the hedged portfolio, and assume that rf = 0, since there is no risk-free asset in the hedged portfolio. We can thus write r p,t = λσp,t 2 + ɛ t. (3.3.23) In order to estimate λ, a GARCH(1, 1)-M (or GARCH in mean) model of the Diagonal Vech GARCH is used: r p,t = λσp,t 2 + ɛ t where ɛ t Ω t 1 N(0, σp,t) 2 (3.3.24) and σp,t 2 = ω + αɛ 2 t 1 + βσp,t 1. 2 (3.3.25) The OHRs vary over time in their article, by using a rolling-window approach. Using the parameters found in the GARCH estimation, the variance-covariance matrix is estimated and the OHRs are compared across utility functions, since this variancecovariance matrix is the same for each utility function. Both Ederington s measure of effectiveness and value at risk (VaR) are computed to compare the hedging strategies.

44 37 3. Review of Futures Markets and Hedging Hsin, Kuo, and Lee (1994); Cecchetti, Cumby, and Figlewski (1988) also base their OHR on the expected-utility maximization, and in addition derive measures of hedging effectiveness based on the same principle. The authors of the former article criticize the OHRs and the corresponding effectiveness measure based on maximizing the Sharpe index (Howard and D Antonio, 1984, e.g.). Hsin, Kuo, and Lee (1994) claim that the Sharpe index is a proper measure to use only when the excess returns are positive and the second order condition of maximization is fulfilled, and can only then be used as an optimization criterion or effectiveness measure. In their paper, a negative exponential utility function with constant ARA is assumed, as well as normal returns. The problem then reduces to one having a mean-variance objective. The OHR is obtained by maximizing the following utility function wrt. w f : V (E(r), σ; λ) = E(r H ) 0.5λσ 2 H, (3.3.26) where E(r H ) = [w S P S E(r S )+w f P f E(r f )]/(w S P S ) is the expected one-period return on the hedged portfolio with w S and w f units of spot and futures respectively, σ H is the standard deviation of the return on the hedged portfolio and λ is the ARA coefficient. This leads to the OHR b = P S P f The new measure of hedging effectiveness is [ ] E(r f ) ρ σ S. (3.3.27) λσf 2 σ f HE = V (E(r H ), σ H ; λ 0 ) V (E(r S ), σ S ; λ 0 ) = V (r ce H, 0; λ 0 ) V (r ce S, 0; λ 0 ) = r ce H r ce S, (3.3.28) where rh ce and rce S are the certainty equivalent returns of the hedged position and the spot position, respectively. This measure incorporates both risk and expected return. The study is done with futures and options from foreign currency data. Junkus and Lee (1985) use the traditional naïve hedge, the MV hedge, a utility maximization hedge and a basis arbitrage model to test the hedging performance of three different stock index futures contracts. All of them are different from the naïve 1:1 hedge, and in most cases the optimal hedge was smaller. The effectiveness of the hedges also varied across the strategies, where the strategy s own criterion gave the best performance in each case.

45 3.3. Optimal Futures Hedge Ratios Dynamic Hedge Ratios It might benefit the hedger to let the hedge ratio vary over time, instead of keeping it fixed throughout the hedging period. As new information gets known to the hedger, he or she might want to update the hedge ratio, such that the risk is minimized at those time points where the hedge is revised. Instead of calculating the hedge ratio based on unconditional moments, it is now based on conditional moments. As written by Chen, Lee, and Shrestha (2003), the MV hedge ratio becomes h 1 Ω t 1 = Cov(R s, R f ) Ω t 1 V ar(r f ) Ω t 1. (3.3.29) What can be easily seen from the dynamic MV hedge ratio, is that the difference between this one and the static is the conditional moments. The dynamic MV hedge ratio can be updated using ARCH and GARCH, both being conditional models, or a moving window method, where the updating of the moments is done by moving the estimation window-wise as new information is known up to the time point where the hedge ratio is calculated. Basak and Chabakauri (2011) provide a solution to the MV dynamic hedge in a general incomplete market setting. A perfect hedge would be possible in a complete and frictionless market through static or dynamic trading. This is rarely the case given market frictions making them incomplete. Instead of deriving a dynamic hedge that is minimized at the beginning of the hedging horizon, and from which a hedger may deviate at a later point in time if wanted, they obtain time-consistent hedges by dynamic programming. When the market considered is complete, the risk-neutral probability measure is unique and can be used consistently. Their work is mainly concerned with option hedging, and the hedges can be determined by the Greeks that measure the sensitivities of the asset or commodity wrt. several variables (e.g. its price, volatility, convexity, the interest rate etc.). Since the market is incomplete in their dynamic setting, they find generalized Greeks, which account for this fact. But not all risk can be eliminated when the market is incomplete. In their paper, the non-tradable asset, the liquid asset and the hedger s wealth all follow wiener processes with deterministic functions, and the goal is to minimize the variance of the hedging error (wrt. the amount invested in the liquid asset given initial wealth), defined as the difference between the non-tradable asset price and the wealth. Duffie and Richardson (1991) deal with mean-variance hedging in continuous time, and the hedger has a future commitment in one asset, and can trade futures

46 39 3. Review of Futures Markets and Hedging on another correlated asset to minimize price risk. The asset prices follow geometric brownian motions (GBMs), such that the prices are positively correlated through the two wiener processes. The hedger s problem is to consider a futures strategy θ to maximize expected utility. θ generates profits or losses of G(θ) t = t 0 θ sdf s at any time t, and total final wealth is a function of which strategy θ is chosen: W (θ) = ks t + G(θ) T, where k is the number of units in the future commitment. The problem is formulated as max E(u[W (θ)]), (3.3.30) θ where u(w) = w cw 2 for some constant c. The strategy θ that maximizes the expression in (3.3.30) can also be found by minimizing E([W (θ) L] 2 ) wrt. θ, where L is some target level of final wealth. The traditional minimum variance objective is solved by minimizing the variance of W (θ) wrt. θ. Some numerical examples are provided, and the continuous-time hedges are compared to those that are fixed. The continuous-time hedges also include discrete-time hedges. Kroner and Sultan (1993) account for the long-term cointegration relationship and the dynamic nature of assets by assuming a bivariate ECM with GARCH errors. The ECM deals with the long-term relationship, whilst the GARCH errors permits the distribution s second moments to change over time. Dynamic hedging is then pursued for foreign currency futures. If an investor has a fixed long position of one unit in the spot market, and is short b units in the futures market, the return on the portfolio is x = s bf, where s and f are changes in the spot and futures prices. But since the distribution of spot and futures prices is time-varying, this relationship should be x t = s t b t f t, where t < t, s t and f t are changes in prices between time t and t and b t is the futures position at time t. The optimal futures position is chosen at each time t by maximizing the expected utility function with the following conditional moment E t [U(x t+1 )] = E t (x t+1 ) γσ 2 t (t t+1 ). (3.3.31) The optimal time-varying hedge ratio that maximizes this utility function is given by b t = E t(f t+1 ) + 2γσ t (s t+1, f t+1 ). (3.3.32) 2γσt 2 (f t+1 ) If futures prices were to be martingales, this would lead to the standard MV hedge b t = σt(s t+1,f t+1 ) σ 2 t (f t+1), but with time-varying conditional moments instead of fixed unconditional moments. b is estimated by using a bivariate GARCH(1,1) ECM model.

47 3.3. Optimal Futures Hedge Ratios 40 The model is used for their dynamic hedging strategy, where rebalancing of the hedge ratio only happens if the utility gains from doing so offsets the losses due to transaction costs. Their results indicate that dynamically updating the hedge ratio given transaction costs of doing so improves hedging performance compared to the classic static hedge. Ankirchner, Dimitroff, Heyne, and Pigorsch (2012) investigate the optimal timevarying hedge ratio when the basis is stationary. Bertus, Godbey, and Hilliard (2009) look at a quite similar problem. Both of these articles will be used heavily in later chapters, and will not be covered to a great extent in this chapter. Investors and hedgers would normally want to hedge their exposures through a multi-period horizon. Howard and D Antonio (1991) are two amongst many that consider a multi-period hedging problem. They assume that the spot prices of the asset being hedged shows autocorrelation, and further states that if the autocorrelation is zero, then a single-period hedge is appropriate, and if it is non-zero, a multi-period hedge should be considered. The spot return process follows an autoregressive model of order 1 (AR(1)), and the percentage change in futures prices is i.i.d. and N(0, σ t ) t. There is thus no autocorrelation in the futures return series. Furthermore, it is assumed that the quantity being hedged grows at the rate g. The multi-period hedge ratio is larger than the single-period hedge ratio, and it increases with the number of periods being hedged. Lien and Luo (1993) formulate the problem in the same way. They divide the hedging horizon into T periods, and minimize the variance of the wealth at the end of the hedging horizon, time T. They assume a cointegration process between spot and futures prices, and use an ECM to estimate the hedge ratios. Again, we follow the structure of Chen, Lee, and Shrestha (2003). N s,t is the spot position at the beginning of t, and N f,t = h t N s,t is the futures position taken at time t. The final wealth at time T is T 1 W T = W 0 + N s,t [S t+1 S t + h t (F t+1 F t )] t=0 T 1 = W 0 + N s,t [ S t+1 + h t F t+1 ]. t=0 (3.3.33) When minimizing the variance of W T, the optimal hedge ratios are given by h t = Cov( S t+1, F t+1 ) T 1 ( ) Ns,i Cov( Ft+1, S i+1 + h i F i+1 ). (3.3.34) V ar( F t+1 ) V ar( F t+1 ) i=t+1 N s,t

48 41 3. Review of Futures Markets and Hedging In this case, the optimal hedge ratio h t will vary over time, and the difference from the static hedge ratio is the second term on the right-hand side. Interesting to notice, is the fact that if changes in the current futures prices are uncorrelated to changes in the future spot prices or future futures prices, then h t will be no different than the static MV hedge ratio. This shows that correlation between the prices of the asset or commodity being hedged and the hedging instrument will affect the optimal hedge ratio. Chen and Tu (2010) continue this trend of multi-period hedging, but considers a mean-variance objective rather than minimum variance. Both the spot and futures prices are exogenous stochastic processes, and the hedger is assumed to short a certain number of futures each period. The T -period optimal hedging strategy consists of a series of hedging ratios {b 0, b 1,... b T 1 }. The proceeds from each period is reinvested at the risk-free rate i and held until the end of period T. Transaction costs are ignored, so the wealth at time T is T 1 W T = W 0 + [ S t+1 b t F t+1 ](1 + i) T t 1. (3.3.35) t=0 By maximizing the expected utility EU 0 (W T ) = E 0 (W T ) γv ar 0 (W T ), where γ is the risk aversion parameter, the series of optimal hedge ratios can be solved by backward induction. This starts in T 1, and continues to time 0. They obtain a closed-form solution to the problem, where b T k is the hedge ratio k = 1, 2,... T : b T k = E T k( F T k+1 ) + 2γ(1 + i) k 1 Cov T k ( S T k+1, F T k+1 ) 2γ(1 + i) k 1 V ar T k ( F T k+1 ) + 2γ T 1 t=t k+1 (1 + i)t t 1 Cov T k ( S t+1 b t F T +1, F T k+1 ). 2γ(1 + i) k 1 V ar T k ( F t k+1 ) (3.3.36) As can be seen, when γ = 1, i = 0 and E T k ( F T k+1 ) = 0, b T k becomes the MV dynamic hedge ratio. The hedge ratio given above is hard to estimate since it depends on future hedge ratios, but it can be reduced to a single-period ratio when there is zero serial correlation between spot and futures price changes, and when second moments of the spot and futures price changes contain GARCH effects (Chen and Tu, 2010). Their next proposition is thus that the hedge ratio is reduced to b T k = E T k( F T k+1 ) + 2γ(1 + i) k 1 Cov T k ( S T k+1, F T k+1 ). (3.3.37) 2γ(1 + i) k 1 V ar T k ( F T k+1 )

49 3.3. Optimal Futures Hedge Ratios 42 In order to estimate the latter hedge ratio and the hedging effectiveness, they employ a bivariate GARCH model to model spot and futures price changes in the two well known indexes S&P-500 and FTSE-100. Rao (2011) also examines multi-period hedging, but considers mean-reverting price processes and unbiased futures markets. He further looks at the optimal hedging strategy when hedging is done using nearby futures contracts that are being rolled over, and when matched-maturity contracts are used. He finds that if both price processes mean-revert at around the same rate and one uses matched-maturity contracts, the optimal hedging strategy is front-loaded, meaning that large hedge positions are entered into long before the hedging horizon is over. If the hedged price process shows higher mean reversion than the futures price process, a single-period hedge seems more appropriate (or back-loaded). When hedging with nearby futures contracts, it depends on the mean reversion of the hedged price process. Neuberger (1999) specifically looks at an agent who has a long-term commitment to supply a certain commodity in the future, and wishes to hedge using futures with a nearby maturity that are being rolled over. He estimates the model parameters through a cross-sectional regression, and the main assumption is that the expected value of the opening price of a futures contract is a linear function of other futures contract prices. Low, Muthuswamy, Sakar, and Terry (2002) consider a multi-period MV hedge based on a cost-of-carry model for the futures prices. Rolling hedges are not considered, so futures contracts maturing the closest (but after) to the end of the hedging horizon are used. The static cost-of-carry outperforms the dynamic one in their article, and both of them outperform the cointegrated ECM hedge (Lien and Luo, 1993), the GARCH hedge (Kroner and Sultan, 1993), and the classic MV hedge (Ederington, 1979). Static hedging is less expensive since transaction costs are lower, and static hedging models are less prone to estimation errors than are dynamic models. Static hedge ratios might therefore outperform the dynamic ones, even though they should seem more appropriate according to theory. In the period prior to the terminal period T, the following hedge ratio is calculated h T 1 = Cov T 1(S T, F T ). (3.3.38) V ar T 1 (F T ) In order to find the optimal hedge ratios in earlier periods, backward recursion is

50 43 3. Review of Futures Markets and Hedging used with the following formula for the optimal hedge ratio h T k = Cov T k[f T k+1, S T T 1 t=t k+1 h t (F t+1 F t )] V ar T k (F T k+1. (3.3.39) 3.4 Proxy Hedging A problem that exists for smaller and less developed commodity spot markets is the non-existence of a corresponding forward or futures market. The hedger must then cross hedge, or proxy hedge 3 (throughout this thesis we will use the latter term). Anderson and Danthine (1981) say that many spot products do not have an obvious futures market, and a proxy hedge can be entered into with a futures position in a related commodity. A futures position is taken in another, but highly correlated, commodity. According to Duffie (1989), the general rule is to choose the futures contract whose price movements have the highest possible correlation with price movements in the spot commodity being hedged. If the two are not correlated, this can even worsen the initial situation. This section will highlight some of the previous work done within the field of proxy or cross hedging. In general, the estimation procedure of the optimal hedge ratio will be similar to that of hedging in markets where a liquid derivative market exists for the commodity being hedged. The ability to hedge is essential for many market participants, them being energy producers or consumers of some sort. Often only imperfect instruments are available, which leads to basis risk. The most prominent example, one that has been extensively studied, is airline companies trying to hedge jet fuel (kerosene) price risk. The market for kerosene futures is illiquid, and the hedge must therefore be done in another correlated market, for example in the crude oil market, where futures are more liquid. The correlation between the price series is thus crucial in order to obtain a good proxy hedge. Oftentimes, however, the correlation depends on the time interval chosen in the analysis (Ankirchner, Dimitroff, Heyne, and Pigorsch, 2012). In their empirical analysis of kerosene and crude oil, the daily correlation between log returns of the two series is 0.52, and the weekly, monthly and yearly log returns have correlation coefficients of 0.72, 0.84 and 0.98, respectively. As they mention, there might be a long-term relationship between the log return series of the two commodities. According to Engle and Granger (1987), two series that are 3 Both the terms proxy and cross hedge are used in the literature

51 3.4. Proxy Hedging 44 both integrated of order 1, meaning that they contain one unit root, are separately non-stationary, but if a linear combination of the two is stationary, they are said to be cointegrated. This can for example be the spread between the two, which is then stationary and varies around a certain long-term level. Lien (1996) gives a brief example of why it is important to consider cointegration in the analysis, since not including it will underestimate the optimal hedge ratio and the strategy is thus suboptimal. Ankirchner and Heyne (2010) study a quadratic hedging problem, where they specifically consider hedging of a contingent claim with basis risk. In their case, the correlation between the illiquid underlying of the contingent claim and the hedging instrument is random. Up until now, it seems like there are some main issues to consider when performing a proxy hedge: correlation, liquidity and long-term relationship Liquidity versus Correlation When considering hedging with linear products, both forwards and futures can be used as hedging tools. Forwards are, as known, traded OTC while futures are traded on an exchange. If trading OTC, forwards can be customized to fit the hedger s needs, and many other derivatives are also possible to hedge with. However, compared to futures traded on an exchange, they are rather illiquid and come with relatively higher bid-ask spreads. Still in the case of the airline company (Adams and Gerner, 2012), there are substantial illiquidity premia when trading OTC to hedge the price risk of kerosene. Additionally, the counterparty risk is greater, since payment from the counterparty is not automatically guaranteed by any exchange. Lastly, there might not be sufficient quantities available to hedge all the risk that needs to be hedged (Nascimento and Powell, 2008). Exchange-traded products are often more liquid, and they do not have the equivalent counterparty risk. But, these futures are standardized and there might not exist a perfect linear hedge. The price movements of the hedged asset must therefore be as much correlated as possible to the commodity underlying the hedging instrument. Nascimento and Powell (2008) therefore state that the basis risk when trading on an exchange, in the case of hedging jet fuel, is higher than that of trading OTC. The proxy hedge problem thus often boils down to choosing between correlation and liquidity, as can be summarized in Figure 3.3. Illiquid forwards from an OTC

52 45 3. Review of Futures Markets and Hedging Figure 3.3: Liquidity versus Correlation dealer might be more correlated with the asset or commodity being hedged, and are therefore placed in the north-west region of the graph, whereas more liquidly traded futures from an exchange might be a worse hedging tool with respect to correlation to the hedged asset or commodity. Another example might be when comparing different futures contracts from an exchange. Hedging illiquid German gasoil with a non-existent futures markets by either trading in for example ARA gasoil futures or Brent crude oil futures is what we will focus on later in this thesis, where both are highly correlated to German gasoil, while Brent crude oil is more liquid Long-Term Relationship Lien (1996) states quite clearly that it is of the utmost importance to incorporate the long-term relationship between spot and futures in statistical modeling. In a previous article (Lien and Luo, 1994), it was also shown how important it is to consider the cointegration between the spot and futures prices, more so than the GARCH price movements considered by many others, and that the hedging performance improves by doing so. In the former article mentioned, he shows that omitting this relationship will lead to a smaller futures position than what is optimal, which reduces the hedging performance. Furthermore, he finds that GARCH effects do not affect this conclusion, and that they only lead to time-varying hedge ratios.

53 3.5. Short Summary Short Summary As is obvious from the the last section, is that there exists many different theories on what is the best method and estimation procedure when it comes to determining the optimal hedge ratio. Section 3.3 gave a rather thorough recap of the research done within the field of finding the optimal hedging ratio, it being static or dynamic. It is intended as learning as well as an introduction to the rest of this thesis. That being said, we will not explore all the ideas and areas of research in this thesis, but the ones we find interesting and in-scope for this type of project. We will thus use elements of Section 3.3 and apply them to the theory of Section 3.4 concerning proxy hedging. In the next chapter, we will look into the world of price modeling, and elements from Chapter 2 will be of great value and use when going through the more technical perspective that will be elaborated in this next chapter.

54 Chapter 4 Modeling of Prices 4.1 Introduction In Chapter 2, we saw that energy spot prices have several characteristics that need to be modeled in order to, as correctly as possible, get good estimates of future spot prices. As Eydeland and Wolyniec (2003) explain, the first we have to think about are exactly these characteristics, such as for example mean reversion, seasonality or jumps. The next step is to choose a model that matches these properties of the historical data. Model parameters can then be estimated, and the performance of the model is checked out-of-sample. If the fit is relatively good, data can be simulated and prices of futures prices can be calculated such as to match that of liquid futures prices in the market. All this sounds quite easy, but in reality it is not. The model should fit even though the market changes, so the estimated parameters should be relatively stable. In addition, spot data is quite hard to come across in energy markets. By using both spot data and futures prices, we have much more data at hand, and a more correct estimation is possible. This will be further explored in Chapter Stochastic Processes Future spot prices are uncertain, and they should be modeled as stochastic processes (Miltersen, 2011). A stochastic process varies over time in a partly random way (Dixit and Pindyck, 1994). Also, a stochastic process can be stationary, meaning that the statistical properties of the process are roughly constant over longer periods, 47

55 4.2. Stochastic Processes 48 or non-stationary, which means that movements not necessarily are within a certain range, and that the process has no limits with respect to where it goes. Furthermore, a stochastic process can be either discrete- or continuous-time, the former being a process that at discrete time points changes value, and the latter one that changes continuously through time, as for example the temperature, but is only observed at discrete time points. It can also be discrete- or continuous-state, where the difference is that the values a process take on are either of the discrete type (countable number of possibilities) or the continuous type (any positive real number) (Tsay, 2010). As in Dixit and Pindyck (1994), one of the most basic examples of a stochastic process is the discrete-time discrete-state random walk. If x t is a random walk, the starting value x 0 is known and with a symmetric probability of 0.5, either x t goes up or down by a certain discrete value (e.g. 1) at certain discrete time points t = 1, 2,.... It is described by the following equation x t = x t 1 + ɛ t, (4.2.1) where ɛ t is the random component of this process with a probability distribution of prob(ɛ t = 1) = prob(ɛ t = 1) = 0.5 (t = 1, 2,... ). The value of the jumps or shocks are independent of each other, and in discrete-time they are called white noise (Tsay, 2010). Since the range of possible values for x t increases with t, the variance increases as well, so the process is non-stationary. The expected value of x t seen from t = 0 is 0 if x 0 = 0, given the symmetric probability distribution. We might generalize this random walk, by for example changing the probabilities of a downward or upward jump, or letting the jumps themselves at each point in time be random, continuous variables. An example of a discrete-time continuousstate stochastic process is the first-order autoregressive process that we mentioned in Chapter 3. The AR(1) process is given by x t = θ 0 + θ 1 x t 1 + ɛ t, (4.2.2) where ɛ t in this case is normally distributed with zero mean and variance of σ 2 ɛ (Tsay, 2010), and 1 < θ 1 < 1. In this process, the lagged variable has some explanatory power of future values. This process, contrary to the previous random walk in Equation (4.2.1), is stationary where the long-term expected value equals θ 0 /(1 θ 1 ). The process has a tendency to revert back to its long-term expected value. What both the AR(1) and random walk processes have in common, is that

56 49 4. Modeling of Prices they are markovian. This means that conditional on x t 1, x t is not correlated with x t i for i > 1 (Tsay, 2010). Stated differently, the distribution of future values of x is independent of the past history of x, such that the only thing affecting future values of x is its current value (Miltersen, 2011) Wiener Process The so-called Wiener process (or Brownian motion) is the basic building block when modeling stochastic processes in continuous time. It is the continuous counterpart to white noise in a discrete setting, and it is defined as follows: (Miltersen, 2011) Definition 1 The process W with a value W t following conditions at time t, is characterized by the 1. W 0 = 0, 2. W t W s N(0, t s) for s < t, 3. W has independent increments on non-overlapping time intervals, such that W t0, W t1 W t0,... W tn W tn 1 are simultaneously independent for 0 t 0 < t 1 < < t n, and 4. W has continuous sample paths such that t W t (ω) is continuous ω Ω. First, changes in W are normally distributed, which means that W t can be negative. When modeling asset or commodity prices, this is not a very likely feature (except electricity prices in certain cases). It also implies that as time goes by, the variance will increase linearly with the time interval (Dixit and Pindyck, 1994). Second, it has independent increments. As was stated above, a random walk is composed of independent increments, so the Wiener process might be seen as a continuous-time version of the random walk. This third point implies that it is markovian, meaning that the current value is what is needed to make a good forecast of future values of W. More specifically, if W is a Wiener process and W is the change in W over the interval t, then W = ɛ t t where ɛt now has a mean of zero and a standard deviation of 1. The random variable ɛ t is serially uncorrelated, such that E[ɛ t ɛ s ] = 0 for t s. This translates into dw t = ɛ t dt (4.2.3)

57 4.2. Stochastic Processes 50 Figure 4.1: A simulated Wiener process in a continuous-time setting. Dividing up the differential gives W t = W t dt + ɛ t dt, (4.2.4) where we clearly see that the Markov property is fulfilled. The variance of the change equals V ar[dw t ] = dt, and thus grows linearly with the time horizon (Dixit and Pindyck, 1994). It is rather uncomplicated to simulate a standard Wiener process in an interval (0, T ) given the equations above. An example of a simulated Wiener process can be seen in Figure 4.1. In general, the standard Wiener process described above is not adequate when modeling commodity prices. First, the process is non-stationary. The concept of mean-reverting commodity prices was investigated in Chapter 2, and a stationary process with prices reverting to their long-term mean would be better and more realistic. The volatility of the standard Wiener process goes to infinity with time. Second, the process allows for negative prices. This is not a good feature to include in a model for commodity prices Generalized Wiener Process The standard Wiener process is not very realistic in terms of modeling prices (Tsay, 2010). A logical extension to the Wiener process W above is to introduce drift- and volatility terms. The so-called Brownian motion with drift X, where X typically can be the price process of an asset or commodity, would look like dx t = µdt + σdw t, (4.2.5)

58 51 4. Modeling of Prices where µ is the drift parameter, and σ the volatility parameter. Again, the discrete changes in X, X t, over t are normally distributed with an expected value of E[ X t ] = µ t and a variance of V ar[ X t ] = σ 2 t. A slightly better way to model prices might be to consider time-varying drift and volatility parameters, such that dx t = µ X (t)dt + σ X (t)dw t. (4.2.6) As in Dixit and Pindyck (1994) and from Definition 1, we know that E[dW t ] = 0, such that E[dX t ] = µ X (t)dt. The variance equals V ar[dx t ] = E[dX 2 t ] (E[dX t ] 2 ), containing elements as (dt) 2 and also of higher orders. for a very small dt, these have little overall effect. V ar[dx t ] = σ 2 X (t)dt. These are ignored, since The variance thus becomes Definition 2 The process X is a generalized Wiener process with the following conditions 1. X 0 = x, 2. X t X s N( t µ s X(u)du, t s σ2 X (u)du) for s < t, 3. X has independent increments on non-overlapping time intervals, such that X t0, X t1 X t0,... X tn X tn 1 are simultaneously independent for 0 t 0 < t 1 < < t n, and 4. W has continuous sample paths such that t X t (ω) is continuous ω Ω. As can be seen, the Markov property is still fulfilled, and if we insert µ X (t) = 0 and σ X (t) = 1 t, X becomes a standard Wiener process. The generalized Wiener process can more formally be written as X t = x + t 0 µ X (u)du + t 0 σ X (u)dw u, (4.2.7) where the last integral is a stochastic integral. Following Iacus (2008), we will state one important property concerning the stochastic integral: Definition 3 If I t (X) = t X 0 sdw s is a stochastic integral, where g(t) is a deterministic function of time, then I X (t) is normally distributed with mean and variance ( T ) E X(s)dW (s) = 0 and 0 ( T ) T (4.2.8) V ar X(s)dW (s) = E[X 2 (t)dt]. 0 0

59 4.2. Stochastic Processes Itô Processes and Itô s Lemma A next step in generalizing the Wiener process is to let both the drift- and volatility terms be stochastic functions. By letting the drift- and volatility parameters be both time- and state-varying, or stochastic, we obtain the following definition of the continuous-time stochastic Itô process (Miltersen, 2011) Definition 4 The general SDE or Itô process, is written as dx t = µ X (X t, t)dt + σ X (X t, t)dw t, (4.2.9) where X 0 = x and µ X and σ X are deterministic functions of X t at time t and t itself. The process is still Markovian. The Itô process given above, which is also called a stochastic differential equation (SDE) or diffusion, is not differentiable (Dixit and Pindyck, 1994). Thus, given the need to often work with functions of processes like that, and thereof the need to differentiate or integrate these functions to obtain any solution to the equation, we need to use some other method than what is normally done for normal differential functions. In this case, we will use Itô s lemma, which basically is a stochastic version of a Taylor series expansion. Assume that X evolves according to dx t = µ X (X t, t)dt + σ X (X t, t)dw t, and that we are dealing with a function f(x t, t) that is twice differentiable in X t and once differentiable in t. The Taylor series expansion looks like df = f f dx + X t dt Or, in integral form 2 f X 2 (dx) f 6 X 3 (dx)3 +. (4.2.10) f(t, X t ) =f(0, X 0 ) t 0 t f t (u, X u )du + t 0 0 f xx (u, X u )(dx u ) 2 +, f x (u, X u )dx u (4.2.11) = f t (t, x), f(t,x) = f x x (t, x) and 2 f(t,x) = f x 2 xx (t, x). When taking the limit of this expression, the higher order terms from (dx) 3 and where f(t,x) t further will go to zero. We can then use this formula to find the solution of stochastic differential equations. Excluding mathematical calculations in between, we have the following definition:

60 53 4. Modeling of Prices Definition 5 If X t evolves according to dx t = µ X (X t, t)dt + σ X (X t, t)dw t, and if we are dealing with a function f(x t, t) that is twice differentiable in X t and once differentiable in t, Itô s lemma gives the first differential of f, df, as df = f f dt + dx t f (dx t X t 2 Xt 2 t ) 2 [ f = t + µ X(X t, t) f + 1 ] X t 2 σ2 X(X t, t) 2 f dt + σ Xt 2 X (X t, t) f dw t, X t (4.2.12) where in the second line, the expression for dx t has been incorporated into the formula. This lemma has some nice properties that will be helpful when solving SDEs Spot Price Models Having reviewed some basic stochastic calculus in lighter detail, it is now time to consider which models may be suitable for modeling prices given their time series characteristics. In the models below, the stochastic price process will be of the continuous-time, continuous-state type Geometric Brownian Motion A special case of the Itô process is the so-called geometric Brownian motion with drift, or simply a geometric Brownian motion (abbreviated GBM), where it is assumed that µ X (X t, t) = µx t and that σ X (X t, t) = σx t. It can be formulated as dx t = µx t dt + σx t dw, (4.3.1) where the percentage change X t /X t is assumed to be normal, as an implication of the result from the Brownian motion with drift. Since the percentage change is equal to the log difference between the prices X t, this means that X t in the case of a geometric Brownian motion would be lognormally distributed. This means that absolute changes in for example asset or commodity prices now are positive, given the lognormal assumption. The log difference of X t is normally distributed with expected value of (µ 0.5σ 2 )dt and a variance of σ 2 t. To summarize, we state the following definition 1 For a more extensive treatment of Stochastic Calculus, see e.g. Tsay (2010)

61 4.3. Spot Price Models 54 Figure 4.2: A simulated GBM process with µ = 0.5 and σ = 0.7 Definition 6 the GBM has the following properties (Miltersen, 2011) 1. The GBM X has positive values for the prices X t, 2. Price returns over non-overlapping time intervals are independent, 3. X is Markovian, and 4. The distribution of future value returns of X is independent of the historical and current value of X. A simulated GBM process can be seen in Figure 4.2. As a concrete example, we will use one that is also considered by Eydeland and Wolyniec (2003), where we assume that the price process is described by the following geometric brownian motion (or SDE) ds t = µ S S t dt + σ S S t dw t, (4.3.2) where S 0 = s, W t is the standard Wiener process, and µ S and σ S are constants. Next, we define Z t = ln S t. Furthermore, S t is a solution to the SDE, f is the function which is twice differentiable in S and once differentiable in t, we can define a stochastic process Z as Z t = f(s t, t), (4.3.3) and by Itô s lemma we can find the SDE of Z. Writing f as a function of X and t, it becomes f(x, t) = ln X. The relevant derivatives are f t = 0, f X = 1 X, 2 f X 2 = 1 X 2. (4.3.4)

62 55 4. Modeling of Prices Putting this into the formula for Itô s lemma gives dz t = (µ S 0.5σ 2 S)dt + σ S dw t, (4.3.5) where Z 0 = ln S 0 and Z t N ((µ S 0.5σS 2)t, σ2 St). Using this expression for the change in dz t between times 0 and t, we can express it as dz t = d ln S t = (µ S 0.5σ 2 S)dt + σ S dw t ln S t = ln S 0 + (µ S 0.5σS)dt 2 + σ S dw t S t = S 0 exp{(µ S 0.5σS)dt 2 + σ S dw t }, (4.3.6) implying that S t is lognormally distributed, such that prices are always non-negative, or equivalently that ln S t is normally distributed with an expected value of S 0 +(µ S 0.5σS 2)t and a variance of σ St. As we defined above, Z t = ln S t, or if reversing this relationship, S t = exp Z t. This means that the spot price is lognormally distributed, and that the conditional expected value and variance are as follows (Eydeland and Wolyniec, 2003) E[S t F ] = S 0 e µ St (4.3.7) V ar[s t F ] = S0e 2 2µ St (e σ2 S t 1). The conditional moments will be conditional on the information known at time 0, and this information can be expressed in terms of a filtration F, which basically is a formal way of explaining how information is revealed through time (Lando and Poulsen, 2006). The famous Black-Scholes formula is actually based on the assumption that asset prices follow GBMs. So in general, the GBM has been heavily used to model asset prices given its characteristics. However, when modeling commodity prices, other features need to be taken into account. This leads us to introduce the following model The 1-factor Model As is often seen with for example commodity prices, they tend to revert back to a certain long-term level (Dixit and Pindyck, 1994). Instead of modeling the spot prices as geometric Brownian motions, it could be beneficial to model them as

63 4.3. Spot Price Models 56 Figure 4.3: A simulated OU process with α = 0.5, σ = 0.7 and κ = 7 mean reverting processes. The simplest of them is the so-called Ornstein-Uhlenbeck process, abbreviated as OU process, developed by Ornstein and Uhlenbeck (1930) which looks like dx = κ(α X)dt + σdw, (4.3.8) where κ is the speed of mean reversion and α is the long-term expected level of X. The larger κ is, the more stable will the process X be around α. For this process, the increments are no longer independent. Figure 4.3 shows how a simulated OU process might look like. It reverts back to its long-term mean, and has a more stationary distribution than for example the standard Wiener process and the GBM. It is possible to find explicit solutions to the OU process by applying Itô s lemma to specific functions of f. Following the same line of extensions as above, we can reformulate the OU process as one that describes the return process. This might be more realistic, as the OU process does not in itself guarantee simulation of positive prices or returns dx = κ(µ X)Xdt + σxdw, (4.3.9) where proportional changes in X are mean-reverting. Schwartz (1997) wrote an article about the stochastic behavior of commodity prices. The first and most basic model that he investigates is the so-called 1-factor model, where the log spot price

64 57 4. Modeling of Prices is assumed to be mean-reverting ds = κ(µ ln S)Sdt + σsdw. (4.3.10) By defining X = ln S, the application of Itô s lemma gives us the same OU-process above for X dx = κ(α X)dt + σdw, where α = µ σ2, κ > 0 measures the speed of mean reversion to the long-term 2κ expected log price α, and dw is the standard Wiener process or Brownian motion. What he further does is calculating the conditional first and second moments under the equivalent martingale measure Q. From that, using the properties of the log-normal distribution, he can calculate the futures prices implied by the model under the risk-neutral measure Gibson-Schwartz 2-factor Model The next mean reverting model is the 2-factor model considered by Schwartz (1997), and a bit earlier developed by Gibson and Schwartz (1990). Instead of letting only the spot price be stochastic, the convenience yield is also considered to be stochastic, and it follows an OU process. The interest rate is assumed to be constant. The two factors are following the joint process ds = (µ δ)sdt + σ S SdW S dδ = κ(α δ)dt + σ δ dw δ, (4.3.11) where the mean reversion is incorporated to the model through the stochastic convenience yield, since the two Wiener processes are positively correlated 2 through dw S dw δ = ρdt. If for example there is a positive shock to the spot price process ds, then there probably is a positive shock to the convenience yield process dδ as well given the positive correlation, making dδ t go up. Then it is likely that the convenience yield δ t will increase, which in the spot price process will lead to a decrease in ds t. This is, shortly put, how the mean reverting mechanism functions. The process is Markovian, since both processes only depend on current values and not historical ones. Defining X = ln S, Itô s lemma gives us the process for the log price dx = (µ δ 0.5σS)dt 2 + σ S dw S. (4.3.12) 2 As described in Chapter 3, the correlation is positive given the inventory hypothesis

65 4.4. Relationship between Spot and Futures Prices 58 Again, writing the processes as risk-neutral, they derive the partial differential equation that futures prices must follow and the corresponding solution of futures prices. Schwartz (1997) also considers a 3-factor model, where the interest rate r also is stochastic. The three corresponding processes are correlated through their Wiener increments. In this case, he derives both forward and futures prices since those are not equal when r is stochastic. All the three models above are estimated and compared using oil and copper futures data. The estimation technique will be encountered in the next chapter, and is thus not explored further in this chapter. He finds that the increased degree of complexity of the 3-factor model is not outweighed by its performance in general, so that the 2-factor model is quite adequate. The 3-factor model does however perform better in the valuation of long-term futures contracts. Other than that, the 2-factor model performs relatively similar to the 3-factor model, and it is easier to work with. The two models also imply approximately the same futures volatilities, and this is because the volatility of the interest rate process in the 3-factor model is of a smaller order of magnitude than the volatilities of the other variables. Both of them outperform the 1-factor model. 4.4 Relationship between Spot and Futures Prices The reason for why it is so important to consider futures prices was in large explained in Chapter 3. The main goal of this section is to set up the mathematical relation for pricing purposes, since it will be of the essence when we introduce our spot price model in the next chapter. Then, we will use our estimated spot prices to price futures contracts. In order to price derivatives, we need to more formally introduce the important Q-measure Risk Neutral Probabilities We have mentioned the so-called Q-measure earlier in Chapter 3, but no mathematical explanation has been given. In this section, more formal insights will be stated, which will be important for the pricing of futures contracts and other derivatives in general. The probability measure that investors and companies have is the physical measure P. The probability measure Q is used for pricing purposes, and it has no relation to what investors or companies believe about the future (Miltersen, 2011).

66 59 4. Modeling of Prices The Q-measure is also called the equivalent martingale measure. The problem with pricing derivatives based on the physical probability measure, is that it is not unique. The expectations an investor or a company form about the future is contingent on their risk profile. As the risk profile will be different between agents, pricing under the physical measure will be more complicated. By assuming that the market is complete, such that any security can be replicated by existing products, we can use these risk-neutral probabilities when pricing derivatives. All investors are risk-neutral under Q, and instead of discounting expected future payoffs at a risk-adjusted rate for each investor, we discount using the risk-free rate r. The Q-measure is related to the assumption of no arbitrage. In a complete market with no arbitrage opportunities, there exists a unique riskneutral measure (Miltersen, 2011). The Q-measure is closely related to the physical P-probabilities, in that if P is positive, then Q is also positive. If P is zero for the occurrence of a given event, then Q is also zero for that same event occurring, making the two measures equivalent (Iacus, 2008). For any process X in the economy, and s and t with s t, Q is an equivalent martingale measure equivalent to P such that X s = E Q s [X t ]. It can be shown that the existence of a martingale measure implies no arbitrage, and the market is also said to be complete when there exists a unique martingale measure (Miltersen, 2011). We can thus price any derivative we want. Let us first recap the probability space that we are working in as (Ω, A, P), where Ω is the sample space and P is the physical probability measure. A is a σ algebra, which means that A is a collection of sets such that i) the empty set is in A; ii) if A A, then the set Ā A and iii) if A 1, A 2,... A, then i=1 A i A. So A forms all the events for which a probability can be assigned. Furthermore, P is said to be a probability measure on (Ω, A). In some cases, it is necessary to switch from one probability measure to another, and this can be done by using the Radon-Nikodym derivative. We then say that Z, for example, is the RN-derivative of Q with respect to P, or more formally Z = dq dp. Following the definition by Iacus (2008), Definition 7 If P and Q are two equivalent measures on the probability space (Ω, A), then a random variable Z such that E[Z] = 1 and Q(A) = Z(ω)dP (ω) A

67 4.4. Relationship between Spot and Futures Prices 60 for every A A. Furthermore, we can write E Q [X T ] = E [ P dq X ] dp T, where the RN-derivative can be interpreted as dq = exp t λ dp 0 sdw s t 0 λ2 sds, where λ is the market price of risk. If we are dealing with continuous time stochastic processes, it is not straightforward how to obtain the value of the process at one specific point in time. In addition, if we change probability measure, is the Wiener process unaffected? Having switched to a new probability measure, the mean of the standard Wiener process has probably changed, and the definition of a Wiener process is no longer fulfilled. This is solved by implementing the following definition Definition 8 If W is a Wiener process under P, then by using Girsanov s theorem, we can define a new Wiener process W under Q as d W t = dw t + λ t dt, (4.4.1) where λ is the market price of risk, as defined by the RN-derivative (Miltersen, 2011). This probability transformation is unique, and we can thus price any derivative we want at any point in time Pricing of Futures Contracts Before we start elaborating on the spot price model we have chosen to work with, we here briefly explain the relationship between the spot price of commodities and futures contracts. From Chapter 3, we know that the best estimator of the future spot price is the current futures price F (t, T ) = E[S T F t ], where F t is the filtration at time t. In order to avoid arbitrage opportunities, this relation must hold. This is not always the case, as the risk aversion of investors might cause the futures price to be both higher or lower than the expected future spot price given the nature of the current markets (Geman, 2005). By working in the risk-neutral word with the Q-probability, we can circumvent this inequality since we do not have to take into account any individual risk premium. Let S t be a stochastic process representing the spot price dynamics of an asset. Assume a constant risk-free interest rate, r > 0, so that the futures price and forward

68 61 4. Modeling of Prices price are equal because of the constant discounting. The explanation below is thus equally relevant for pricing of forwards as it is for pricing of futures contracts. Given that F (t, T ) is the futures price at time t T, the payoff from the futures contract maturing in time T, entered at time t, is S(T ) F (t, T ). As we know that the discounted value of the futures contract at time t under the risk-neutral measure Q is a martingale, it must be equal to the discounted expected value of the futures contract at time T. And given that a there is no cost to enter a futures, we have e r(t t) E Q t [S(T ) F (t, T )] = 0, where E Q t represents the expectation with respect to Q. With this we have the formula for the futures price F (t, T ) = E Q t [S(T )]. (4.4.2) The futures price is therefore the expected spot price under the risk-neutral measure Q. As a concrete example, we will below explain how the futures price is estimated assuming a two-factor model according to Gibson and Schwartz (Gibson and Schwartz, 1990; Schwartz, 1997). The spot price, assuming it has a lognormal distribution, is modeled under Q as ds = (r δ)sdt + σ S SdWS dδ = [κ(α δ) λ]dt + σ δ dwδ, where dws dw δ = ρdt and λ is the market price of convenience yield risk (Schwartz, 1997). Accordingly, futures prices must satisfy the following PDE 1 2 σ2 SS 2 F SS + σ S σ δ ρsf Sδ σ2 δf δδ + (r δ)sf S + (κ(α δ) λ)f δ F T = 0 (4.4.3) subject to the boundary condition F (S, δ, 0) = S. Since the spot price is assumed to follow a lognormal distribution, and since the spot price and the convenience yield are jointly normally distributed, parameters for the transition density can be found (Erb, Lüthi, and Otziger, 2010) under P. To find the equivalent parameters under

69 4.4. Relationship between Spot and Futures Prices 62 Q, we need to set µ = r and α = α = α λ/κ. The futures price under Q in t maturing at time T, which is the solution to the PDE above, is given by 3 with F (S, δ, T ) = E Q t (S T ) = S t exp (A(T ) 1 ) e κt δ t, (4.4.4) κ ( A(T ) = + σδ 2 κ σ Sσ δ ρ 2 κ r α ( κ α + σ S σ δ ρ σ2 δ κ ) T e 2κT 4 σ2 δ ) 1 e κt κ 2. κ 3 (4.4.5) Based on this procedure, we can move on to our spot price model and how we consequently estimated the futures prices in this model. As we will see, the futures price is rather similar to the one we just wrote from the two-factor model. 3 For a detailed demonstration, please see Erb, Lüthi, and Otziger (2010)

70 Chapter 5 The 3-factor Spot Price Model Having looked at different spot price models in the previous chapter, we decided on which model to choose for this analysis. Since we are dealing with a proxy hedging problem, some further considerations are needed in order to obtain as good results as possible. This chapter will first begin with a description of our data. Based on the characteristics of the data, we will introduce the 3-factor model that we have chosen as to match our data. This chapter is organized as follows. The first section introduces the concrete problem we are facing, and what important characteristics should be taken into account to appropriately model our strategy. The second section gives an overview of our data. The third section mathematically describes the model. The fourth section goes into detail about how we estimate the model parameters, and the methods we have used to do so. Section five states the results of the estimations, and in the last and sixth section these results will be discussed. 5.1 Rationale How should firms hedge when there is no perfect hedging instrument at hand? This question can be answered in several ways, and for this thesis we have chosen to focus on two rather specific studies. The first article by Ankirchner, Dimitroff, Heyne, and Pigorsch (2012), which we focus on is rather used as input to our model, and demonstrates how to optimally cross hedge when the spread between the futures used for hedging and the risky commodity is stationary. This spread, or so-called basis, becomes a source of risk since the underlying asset is not perfectly correlated 63

71 5.2. Data 64 to the futures contract. It is therefore of the essence to choose a futures contract that is highly correlated to the risky commodity. Given their findings of a strong positive long-term correlation between the two, they assume that the prices are cointegrated. This is also supported by empirical tests. Instead of imposing a cointegrated vector, they assume that the spread between the log prices is stationary. The long-term relationship is therefore an essential part of their analysis, which allows for inclusion of the basis-risk which varies in a stationary way over time. This is modeled as an OU-process. While their futures price is modeled as a GBM, we chose another path. The second, and mainly contributing, article we focus on is one by Bertus, Godbey, and Hilliard (2009). The main motivation of this article is also how to best strategize a cross hedge. They derive minimum variance hedges by assuming that the commodity price moves according to the two-factor model (Gibson and Schwartz, 1990), and that the log spread moves as a mean reverting OU-process. It is therefore a jointly estimated 3-factor model. They specifically focus on the airline industry, and cross hedge jet fuel risk with crude oil futures given their high positive correlation. Our concrete problem is a large energy corporation that wants to hedge away risk in German gasoil. There is no liquid German futures market for gasoil, hence the need to hedge with another commodity. For this purpose, we have chosen two potential hedges: ARA gasoil and Brent crude oil futures. We will for both of these look at different hedging strategies, and finally compare their effectiveness. 5.2 Data Throughout this thesis, we have emphasized the importance of cross hedging with a highly correlated futures contract. We have chosen to work with monthly price data, which ranges from February 1, 2005 to December 1, This gives us time-series comprising 95 prices. We further have an equal length of data on twelve different futures contracts on both Brent crude oil and ARA gasoil. The futures contracts are generic and range from front-month to twelve months ahead. The ICE Brent Futures is a deliverable contract based on EFP delivery with the option to cash settle, with Dated Brent as underlying. ARA gasoil futures is based on the underlying heating oil barges delivered in ARA. It has a merchantable quality at a density of 0.845

72 65 5. The 3-factor Spot Price Model Figure 5.1: Spot Prices in EUR/mt kg/litre in vacuum at 15 degrees Celsius 1. All the prices are quoted in euros per metric tonnes 2, after the conversion we did. First, we downloaded all the price data in Euros, and not in U.S. Dollars which is the standard currency. Futures prices of Brent crude oil was denominated in Euros per barrel. ARA gasoil was denominated in Euros per metric tonnes. We wanted all our data in Euros per metric tonnes, and to do that we used conversion factors from BP. 1 barrel equals tonnes. In addition, 1 hecto litre equals tonnes. To convert Brent crude from EUR/barrel to EUR/tonnes, we thus divided by Since ARA gasoil was denominated in EUR/tonnes, but with a deliverable density quality of kg/hl, we needed to adjust the German gasoil prices to match this, and thus multiply these prices by As can be seen from Figure 5.1 all three prices series move quite similarly over time. From Figure 5.1 it is not specifically clear whether the spot prices are mean reverting or not. They also do not seem to follow a GBM, as the variation in prices is limited to a certain interval. They do seem to revert to a long-term level of around 400 to 600 EUR/mt. The spot prices do not show any clear seasonal patterns. The reason for this might be that we are operating with monthly data. We are choosing to model the spot prices as mean reverting, because we want the model to fit any time period, and any frequency of prices. It is a well known phenomenon that 1 Taken from Bloomberg when downloading the data 2 Conversion rates can be found in Appendix A.2, Table A.1, and are taken from

73 5.2. Data 66 many energy prices are in fact mean reverting, and do not generally follow a GBM. Summary statistics for the spot prices are reported in Table 5.1. As can be seen from Table 5.1, German gasoil spot prices are on average higher than that for Dated Brent and ARA gasoil, and they also have a higher volatility. Statistic Dated Brent ARA gasoil German gasoil N. of Obs Min Max Mean SD Table 5.1: Summary Statistics for Spot Prices Figure 5.2 shows historical prices of both Brent and ARA gasoil futures. We have chosen to focus graphically and statistically on certain futures contracts for the sake of brevity and unnecessary repetition. Figure 5.2 thus shows front-month, 3- months, 6-months, 9-months and 12-months futures for each of the two commodities. As becomes obvious by looking at Figure 5.2 is how similarly the different futures prices behave. Summary statistics for the different futures prices are given in Table 5.2. Statistic Brent 1M Brent 6M Brent 12M GO 1M GO 6M GO 12M N. of Obs Min Max Mean SD Table 5.2: Summary Statistics for Brent and ARA gasoil (GO) Futures Prices When looking at optimal hedge ratios further on, correlation coefficients between German gasoil prices and the different futures prices will become important. We have mentioned that correlation is the most important criteria for finding a suitable cross hedge, and this will be discussed in more detail in the following chapter. Table 5.3 shows the correlation coefficients between German gasoil and the different futures

74 67 5. The 3-factor Spot Price Model (a) Brent Futures Prices (b) ARA gasoil Futures Prices Figure 5.2: Brent and ARA gasoil Futures Prices in EUR/mt prices. The correlation coefficients range from to 0.975, indicating that the correlation is high even for the lowest values. It is generally higher between German gasoil and ARA gasoil futures than it is for German gasoil and Brent futures. What is interesting further, is how the log spread evolves between the illiquid German gasoil and the more liquid futures prices mentioned above. Figure 5.3 shows the log spreads between some chosen futures contracts for each of the two commodities 3. Not surprisingly, the spread between German gasoil and ARA gasoil futures is a bit more stable than that between German gasoil and Brent crude oil 3 Graphs of all log spreads can be found in Appendix A.1, Figures A.1 and A.2

75 5.2. Data 68 Correlation Coefficients between GGO and Futures Prices GGO GGO GGO GGO BR 1M BR 7M GO 1M GO 7M BR 2M BR 8M GO 2M GO 8M BR 3M BR 9M GO 3M GO 9M BR 4M BR 10M GO 4M GO 10M BR 5M BR 11M GO 5M GO 11M BR 6M BR 12M GO 6M GO 12M Table 5.3: Correlation Coefficients between German gasoil and Brent (BR) and ARA gasoil (GO) Futures Prices futures. For both Brent and ARA gasoil, the stationarity also seems to weaken as time to maturity increases. This makes perfect sense as front-month futures contracts are often very similar in its characteristics as the spot price underlying the futures. In this particular case, since the correlation is so high between the different prices, see Table 5.3, the same logic appears to be true. Statistic GGO and Brent 1M GGO and Brent 6M GGO and Brent 12M PP *** *** *** (0.010) (0.010) (0.010) ADF *** ** (0.021) (0.239) (0.437) Statistic GGO and GO 1M GGO and GO 6M GGO and GO 12M PP *** *** *** (0.010) (0.010) (0.010) ADF *** ** * (<0.010) (0.057) (0.333) Table 5.4: Stationarity Tests of the Log Spreads. P-Values are reported in parenthesis. ***/**/* means rejection of unit root at a 1 %/5%/10% significance level, respectively. In order to test for stationarity, ADF-tests have been performed for all possible spreads. In Table 5.4, the results from the ADF-tests of some chosen spreads are reported. We have performed two different tests for stationarity, the well-known Augmented Dickey-Fuller test and the Phillips-Perron test. Both of them checks

76 69 5. The 3-factor Spot Price Model (a) GGO and BR1M (b) GGO and BR6M (c) GGO and BR12M (d) GGO and GO1M (e) GGO and GO6M (f) GGO and GO12M Figure 5.3: Log Spreads between German gasoil and Brent and ARA gasoil Futures. These six graphs show the log spreads between the illiquid German gasoil and front-month, 6-months and 12-months Brent crude oil and ARA gasoil futures contracts whether there exists a unit root, and the existence of a unit root is rejected if the ADF or PP-statistic is more negative than -3.5 (1%), -2.9 (5%) and -2.6 (10%), where the numbers in parenthesis show the statistical significance. When a unit root is rejected, the time series is stationary. The Phillips-Perron (PP) unit root test is different from the ADF test, primarily because of how it deals with serial correlation and heteroskedasticity in the errors. Two advantages of the PP test are that it is robust to general forms of heteroskedasticity in the error term, and that one does not have to specify a lag length for the test regression. As can be seen from Table 5.4, German gasoil and ARA gasoil futures are more stationary in their log spreads than German gasoil and Brent are. There are also some differing evidence with respect to the two tests, but both of them do show some evidence of stationarity, at least in the closer time to maturity futures contracts. This is also what is shown graphically in Figure 5.3.

77 5.3. The 3-Factor Model The 3-Factor Model The model we consider is regarding a value-maximizing firm with a fixed exposure to the German gasoil spot price at a fixed future date. Since liquid German gasoil futures contracts do not exist, we must hedge in another but related market General Setup In the line of what Bertus, Godbey, and Hilliard (2009) explain, the firm considered is committed to buying a fixed amount, m, of German gasoil in a future period t at spot price. After having chosen a hedging strategy, the firm s cash flow is V t = m GGO t + h 0 F 0, (5.3.1) where m is the number of units of the German spot that is bought at time t, GGO t is the spot price of German gasoil at time t, h 0 is the number of futures contracts entered today, at time 0, and F 0 = F t,t F 0,T, where F 0,T is the price today of the futures contract used as hedging instrument which matures at time T, and F t,t is the futures price at time t. As we have explained earlier, we want to model the log spread between German gasoil and the futures used as hedging instrument as stationary or mean reverting. The log spread is thus given by b t = ln(ggo t /F t ), and can further we written as GGO t = F t e bt, (5.3.2) where GGO t is the spot price of German gasoil at time t, F t is the futures price at time t of a given hedging instrument maturing at a given time T, and b t is the log spread. Furthermore, we suppose that the log spread is mean reverting and stationary. The reason for this is that it accounts for the long term cointegration relationship between German gasoil and the highly correlated futures. We therefore let the spread follow the mean reverting OU process db t = κ b (α b b t )dt + σ b dw b (t), (5.3.3) where κ b is the speed of mean reversion, α b is the long term spread level, σ b is the volatility of the spread, and W b (t) is the standard Wiener process of this SDE. When modeling the spot price of the hedging instrument, it being either Brent or ARA gasoil, we also want to take into account a stochastic convenience yield.

78 71 5. The 3-factor Spot Price Model We thus model the spot price to follow a two-factor mean-reverting model, as do Gibson and Schwartz (1990) ds t = (µ δ t )S t dt + σ S S t dw S (t) dδ t = κ δ (α δ δ t )dt + σ δ dw δ (t), (5.3.4) where δ is the convenience yield, µ is the mean reversion level of the spot return series, σ S and σ δ are the volatilities of the spot and convenience yield respectively, κ δ is the speed of mean reversion of the convenience yield, α δ is the long term mean reversion level of the convenience yield, and W S (t) and W δ (t) are the standard Wiener process increments of the spot and convenience yield processes respectively. These three stochastic differential equations are correlated through the Wiener processes, and their correlation coefficients are ρ S,δ, ρ S,b and ρ b,δ. There are in effect three factors that need to be taken into account when calibrating the model; the log spread b t, the spot price S t and the convenience yield δ t. We have the following system that needs to be estimated ds t S t (µ δ t ) dδ t = κ δ (α δ δ t ) db t κ b (α b b t ) dt + S t σ S dw S σ δ (αdw S + βdw δ ) σ b (γdw S + θdw δ + vdw b ), (5.3.5) and this is easily solved by performing a Cholesky decomposition. We will go through this step by step Cholesky Decomposition of Correlations Let us first look at the covariance between S and δ. We know that they are only correlated through the Wiener processes, so that we can write Cov(S, δ) = Cov(Sσ S W S, ασ δ W S + βσ δ W δ ) = Cov(Sσ S W S, ασ δ W S ) + Cov(Sσ S W S, βσ δ W δ ). Since the second term on the right-hand side is zero, because by definition the the two different Wiener processes are independent, this gives us Cov(S, δ) = Cov(Sσ S W S, ασ δ W S ) = Sσ S σ δ α, 4 Please note that α used in the Cholesky decomposition is not the same as α δ and α b

79 5.3. The 3-Factor Model 72 such that we can write α = Corr(S, δ). (5.3.6) By doing the same kind of manipulation with the covariance between S and b, we find that γ = Corr(S, b). (5.3.7) Finally, regarding the covariance between δ and b, we have Cov(δ, b) = Cov(ασ δ W S + βσ δ W δ, γσ b W S + θσ b W δ + vσ b W b ) = Cov(ασ δ W S, γσ b W S ) + Cov(ασ δ W S, θσ b W δ ) + Cov(ασ δ W S, vσ b W b ) + Cov(βσ δ W δ, γσ b W S ) + Cov(βσ δ W δ, θσ b W δ ) + Cov(βσ δ W δ, vσ b W b ), and by canceling all the covariance terms that include independent Wiener increments, we are left with Cov(δ, b) = Cov(ασ δ W S, γσ b W S ) + Cov(βσ δ W δ, θσ b W δ ) = αγσ δ σ b + βθσ δ σ b = (αγ + βθ)(σ δ σ b ), such that we finally can write αγ + βθ = Corr(δ, b). (5.3.8) Now we have a system looking like this α = ρ S,δ γ = ρ S,b αγ + βθ = ρ δ,b, (5.3.9) so we need some additional rules to being able to use it in our estimation further on. The following two conditions holds true (Miltersen and Schwartz, 1998): α 2 + β 2 = 1 γ 2 + θ 2 + v 2 = 1. (5.3.10) Using the first equation in system (5.3.9) and the first condition in (5.3.10), this implies that α = ρ S,δ β = 1 ρ 2S,δ. (5.3.11)

80 73 5. The 3-factor Spot Price Model Now, using the equations from (5.3.9), we find that ρ S,δ ρ S,b + βθ = ρ δ,b βθ = ρ δ,b ρ S,δ ρ S,b θ = ρ δ,b ρ S,δ ρ S,b, 1 ρ 2 S,δ and by using the second equation in (5.3.10) with this new information, we get ρ 2 S,b + γ 2 + θ 2 + v 2 = 1 ρ δ,b ρ S,δ ρ S,b 1 ρ 2 S,δ 2 + v 2 = 1, which can be written as 2 v = 1 ρ 2 S,b ρ δ,b ρ S,δ ρ S,b. (5.3.12) 1 ρ 2 S,δ We have then finally found all the parameters α, β, γ, θ and v in terms of the correlation coefficients. To summarize, they are given by α = ρ S,δ β = 1 ρ 2 S,δ γ = ρ S,b θ = ρ δ,b ρ S,δ ρ S,b 1 ρ 2 S,δ v = 1 ρ 2 S,b ρ δ,b ρ S,δ ρ S,b. 1 ρ 2 S,δ (5.3.13) System of Equations to be Simulated We have all the elements we need to find our final system that needs to be simulated. We want to simulate all the three processes, namely the spot price, the convenience yield and the log spread. As explained by Erb, Lüthi, and Otziger (2010), the logarithmic spot price S t and the convenience yield δ t are jointly normally distributed, with a transition density that equals

81 5.3. The 3-Factor Model 74 ( ln S δ t ) (( µln S (dt) N µ δ (dt) ) ( )) σ 2, ln S (dt) σ ln Sδ (dt) σ ln Sδ (dt) σδ 2(dt). (5.3.14) We will for the sake of brevity use the results from Erb, Lüthi, and Otziger (2010), and not derive the transition density ourselves. The transition density has the following moments µ ln S (dt) = ln S 0 + (µ 12 ) σ2s α δ dt + (α δ δ 0 ) 1 e κ δt µ δ (dt) = e κδt δ 0 + α δ (1 e κδt ) ( σln 2 S(dt) = σ2 δ 1 (1 e 2κδdt ) 2 ) (1 e κδdt ) + dt κ 2 δ 2κ δ κ δ + 2 σ ( Sσ δ ρ S,δ 1 e κ δ ) dt dt + σ κ δ κ Sdt 2 δ κ δ (5.3.15) σδ(dt) 2 = σ2 δ (1 e 2κδdt ) 2κ δ σ ln S,δ (dt) = 1 {( ) } σ S σ δ ρ S,δ σ2 δ (1 e κδdt ) + σ2 δ (1 e 2κδdt ). κ δ κ δ 2κ δ Using these moments, we can derive the processes we want to simulate. The spot price process looks like S t = S t 1 + (µ 12 ) σ2s α δ + (α δ δ t 1 ) 1 e κ δdt κ δ ( σδ (1 e κ 2 δ 2κ 2κ δdt ) 2 ) (1 e δ κ κ δdt ) + dt δ + 2 σ ( ) Sσ δ ρ S,δ 1 e κ δ dt dt + σs 2 κ δ κ dt dw S. δ (5.3.16) The process for the convenience yield looks like δ t = α δ (α δ δ t 1 )e κ δdt + σ δ 1 e 2κ δdt And the process for the log spread looks like 2κ δ (αdw S + βdw δ ). (5.3.17) b t = α b (α b b t 1 )e κ bdt 1 e + σ 2κ bdt b [(γ + αθ)dw S + βθdw δ + vdw b ]. 2κ b (5.3.18)

82 75 5. The 3-factor Spot Price Model It is worth mentioning that the simulations of spot, spread and convenience yield are done under the physical measure P. As Ankirchner, Dimitroff, Heyne, and Pigorsch (2012) emphasize, the physical measure is the only relevant measure when doing analyses on hedging and risk management. Using the spot price and the convenience yield, we get the expression for the futures price (Bertus, Godbey, and Hilliard, 2009). Since we are assuming a constant and positive risk-free interest rate, the futures price coincides with the forward price. The futures price using the 3-factor model is F (S 0, δ 0, T ) = S 0 A(T )e rt H δ(t )δ 0, (5.3.19) where ( (Hδ (T ) T )(κ 2 δ A(T ) = exp α δ κ δ λ δ σ δ 0.5σδ 2 + ρ S,δσ S σ δ κ δ ) and H δ (T ) = (1 e κ δt )/κ δ (Bertus, Godbey, and Hilliard, 2009). κ 2 δ σ2 δ H2 δ (T ) ), 4κ δ Looking at the expression for the futures price one sees that by calculating it one obtains the price at time t = 0 for a futures contract with maturity T. Thus it is reasonable to say that one could essentially substitute T with T 0, where 0 represents t = 0. If we then continue this reasoning one can replace all t = 0 with t = t, thus obtaining an expression for futures prices for all time steps up to maturity T, meaning one discounts the price forward. In this case the expression becomes F (S t, δ t, T t) = S t A(T t)e r(t t) H δ(t t)δ t. (5.3.20) 5.4 State-Space Models and the Kalman Filter We developed in the previous section the mathematical basis for what is needed to estimate our wanted parameters. However, state variables of these models are often not easily observable. In many cases, spot prices are so hard to come by that the nearest to maturity futures contract is used as a proxy for the spot price (Schwartz, 1997). This also applies to the convenience yield, which is practically unobservable. Futures contracts are however traded frequently, and we can easily observe futures prices. Our spot price model assumes Markovian prices. When this is the case, and we have rather limited data on the spot prices, we can put our model in the state space

83 5.4. State-Space Models and the Kalman Filter 76 form. This is a good way of handling missing values (Tsay, 2010). Once the model is in state space form, we can apply the Kalman filter approach. The Kalman filtering method has been used extensively in the field of engineering, but has become widely used in finance and economics more recently (Arnold, Bertus, and Godbey, 2008). It has been helpful in estimating unobservable parameters and state variables in commodity prices. This filtering approach updates knowledge of the state variable each time new data becomes available. We can then estimate the parameters of the model, and induce time series of the unobservable spot price. The reason that it is called filtering, is because as new information becomes known, more noise is filtered out and estimates become more accurate. Following the relatively intuitive explanation of Arnold, Bertus, and Godbey (2008), the Kalman filter method consists of a measurement equation, which relates the unobserved variable to the observable one, and a transition equation, which allows the unobserved variable to change over time. The algorithm basically goes like this: 1. Predict the future unobserved variable in t + 1 based on its current estimated value in t 2. Use the predicted unobserved variable to predict the future observable variable in t In t + 1, when the observable variable actually occurs, calculate the prediction error 4. And generate a more accurate estimate of the unobserved variable at t Repeat this process over again for t + 2 etc. As Bertus, Godbey, and Hilliard (2009) do, we denote the front-month futures contract as F 1, and its time to maturity as T 1. The same notation goes for second month, third month,..., and twelve month futures contracts. The measurement equation is y t = Z t α t + c t + ɛ t, t = 1,... T (5.4.1)

84 77 5. The 3-factor Spot Price Model where ( c t = ( y t = ln(f 1 t ) ln(f 2 t )... ln(f 12 t ) ln(ggo t ) 1 H(T 1) 0 1 H(T 2) 0 Z t =... 1 H(T 12) ) 13 1 ln(a(t 1)) + rt 1 ln(a(t 2)) + rt 2... ln(a(t 12)) + rt 12 0 ( ) α t = ln(s t ) δ t b t 3 1 ) 13 1 where ɛ t is a 3 1 vector of uncorrelated disturbances with E[ɛ t ] = 0 and V ar[ɛ t ] = GG t, and GG t is a matrix with measurement errors. The different dimensions are shown as subtext behind each vector or matrix. For the transposed vectors, the dimension in shown in their transposed form. with The transition equation looks like α t = T t α t 1 + d t + v t, (5.4.2) 1 t 0 T t = 0 1 κ δ t κ b t 3 3 ( d t = (µ 0.5σS 2) t κ δα δ t κ b α b t where v t is the noise term with E[v t ] = 0 and V ar[v t ] = HH t. HH t is described by the following matrix σs 2 t ρ S,δσ S σ δ t ρ S,b σ S σ b t HH t = ρ S,δ σ S σ δ t σδ 2 t ρ b,δσ b σ δ t ρ S,b σ S σ b t ρ b,δ σ b σ δ t σb 2 t In Appendix B.4, the entire code for performing the Kalman filter estimation in R is shown. Now that we have the system of equations that we need in order to estimate, how are the parameters then estimated? We assume that the distribution for each of )

85 5.5. Estimation and Results 78 the predicted observable variables is serially independent and normally distributed, and the log likelihood function will then be maximized (Arnold, Bertus, and Godbey, 2008), such that the probability of the observable data actually occurring is at its highest. From this log likelihood function the observable data will then become known, and the parameters we want to estimate are chosen so that they maximize the maximum log likelihood value function. Often a log-likelihood function is used to simplify calculations. When the parameters we want are estimated, the Kalman filtering is done over again. This produces new time series estimates, and the likelihood function is then maximized again by producing new parameters that do exactly that. This continues until the maximum likelihood function is at its best and does not improve significantly if more runs are performed. 5.5 Estimation and Results The parameters we want to estimate are µ, σ S, κ δ, α δ, σ δ, κ b, α b, σ b, ρ S,δ, ρ S,b, ρ δ,b, λ δ, and the measurement errors in the GG t matrix, ME and ME13. The estimation results are shown in table 5.5. We have done the estimations using both Brent and ARA gasoil futures as separate inputs, and for two different time periods. We have also chosen to work with a risk-free interest rate of 5%. The reason we chose exactly this number is after having looked at historical Treasury Rates in the U.S. Some might argue that it is a bit high give today s interest rate level, but given the past history we chose this rate of 5%. As can be seen from Table 5.5, most of the parameters are quite significant. We ran the Kalman filter several times, and changed our initial variables for each run, and the results were quite stable around the same level. The estimated parameters also match some general views about the original price series. ARA gasoil is situated on a slightly higher price level, and also shows higher volatility. This matches our findings. What is rather remarkable with these estimated parameters, are the high volatility parameters for the spot price processes. When using Brent crude oil futures, the volatility parameter is higher when excluding the last year from the data in the estimation. The opposite holds true when using ARA gasoil futures. When looking at Figure 5.2 (a), it seems as tough the futures prices are rising sharply in December 2011, whilst they are rather stable in Figure 5.2 (b). It also seems as though the prices vary more for Brent this last year from Dec to Dec. 2012

86 79 5. The 3-factor Spot Price Model than for ARA gasoil. Leaving this year out of the estimation might very well lead to a higher volatility for the Brent estimation than for the ARA gasoil estimation. We did also try to run the estimation through the Kalman filter several times, which all lead to rather high volatility estimates for the spot price processes. Again, looking at Figure 5.2, we see a large downward jump in both Figures (a) and (b) in It is of no news that the sharp decline in futures and spot prices at that time was due to the economic climate and the financial crisis. Given that our data set is rather short (ranging only from 2005 to 2012) with monthly observations, a large downward jump like that makes our data sets very varying. This might also be a reason for the high volatility estimates. We also notice that the speed of mean reversion for the log spread process, κ b is increasing when using Brent crude oil, and when comparing the longer data set used with the shorter one. The opposite again holds for the results using ARA gasoil. This might be directly related to the volatility parameters. In the case where we have used Brent futures, when futures prices are more volatile, this leads to a higher volatility estimate for the spot price process, and thus higher volatility in the simulated futures prices which are based on the estimated spot price parameters. Since the log spread process is mean reverting, a higher volatility in the prices requires a higher speed of mean reversion to lead the log spread back to its long-term mean reversion level. The volatility parameter for the log spread process is then lowered when using the shorter data set. As can be seen from Table 5.5, the opposite holds for Panel B, where we have used ARA gasoil futures for the estimation. The reliability of our results will be discussed in the concluding section of this chapter, Section 5.7, in relation to the model as a whole.

87 5.5. Estimation and Results 80 Feb. 1, 2005 to Dec. 1, 2012 Feb. 1, 2005 to Dec. 1, 2011 Panel A: Brent crude oil Futures µ (0.029) (0.021) σ S (0.009) (0.013) κ δ (0.048) (0.095) α δ (0.002) (0.001) σ δ (0.019) (0.032) κ b (2.489) (2.158) α b (0.023) (0.021) σ b (0.176) (0.111) ρ S,δ (0.014) (0.013) ρ S,b (0.031) (0.025) ρ δ,b (0.046) (0.047) λ δ (0.024) (0.026) ME ( ) ( ) ME (0.028) (0.020) Panel B: ARA gasoil Futures µ (0.031) (0.043) σ S (0.002) (0.343) κ δ (0.137) (0.124) α δ (0.004) (0.006) σ δ (0.008) (0.042) κ b (0.783) (2.808) α b (0.027) (0.040) σ b (0.036) (0.183) ρ S,δ (0.025) (0.051) ρ S,b (0.132) (0.181) ρ δ,b (0.053) (0.057) λ δ (0.043) (0.022) ME ( ) ( ) ME (0.001) (0.074) Table 5.5: Kalman Filter Parameter Estimates of the 3-factor Model. Standard errors are reported in parenthesis, and Panel A shows the estimated parameters using Brent futures, and Panel B the results using ARA gasoil futures.the first column with results are estimated using the entire data set, whilst the third and last columns shows results based on the entire data set excluding the last year

88 81 5. The 3-factor Spot Price Model 5.6 Simulation of Prices Having found the parameters for both Brent crude oil and ARA gasoil, we simulated spot prices for each product. 10,000 paths were simulated for the spot price and the spread between the illiquid German gasoil and futures prices, and for each of the products. The simulations were done based on equations (5.3.16), (5.3.17) and (5.3.18) from Section Even though the processes used for the simulations are of the continuous-time type, the simulations are essentially done discretely, one time step at the time. So as to match our monthly historical prices, the time step used was monthly as well. The way of doing this is setting dt = 1/12. Each simulation was done 12-months into the future. The argument behind the amount of simulations is that it is important to have enough simulations so as to take into account the great volatilities in energy prices. This essentially is a Monte Carlo method of simulating spot prices. Options that are path-dependent, meaning not only dependent on the terminal price, are often priced using a Monte Carlo method. This entails simulating a large number of paths that the option price may take to reach the terminal price and then taking the discounted average of the option value over all these paths (Benninga, 2008). The rationale behind this is that the more paths simulated the more accurate the option value will be. The same reasoning is used in our model; simulating enough price- and spread paths so as to predict the actual prices. However, in our simulations we do not take the average of the prices and end up with one price path, rather we continue our analysis by also calculating 10,000 cash flows at a certain point in time and then taking the standard deviations of these. A more in-depth explanation of this will be presented in Chapter 6. Furthermore, as our parameters is based on a dataset consisting of prices on futures contracts with maturities up to 12-months, thus to match this dataset, the spot prices and spread were simulated 12 months ahead. As mentioned in the previous section parameters were found for two time periods. One of the time periods corresponded to the whole dataset. When simulating for the whole dataset, we assumed that time 0 was the last date of the dataset, thus the spot price at time 0 is the last price in the dataset. In this way the 10,000 trajectories are essentially predicting the spot prices 12 months ahead, continuing

89 5.6. Simulation of Prices 82 the historical prices. The second set of parameters is based on a time period corresponding to the whole dataset excluding the last year. This results in an out of sample prediction, giving the possibility of checking the model s fit against the data. The simulation thus starts with the last spot price of the shorter dataset, and predicting spot prices then can be compared to actual data. Figure 5.4 illustrates this out of sample prediction (black trajectories) against the actual data (red trajectory). Only 20 trajectories are shown as there would not be room for all of them. Looking at the figure it seems as though the model with the estimated parameters fits relatively well. The trajectories seem to evolve around the same mean as the actual data, even though the volatility of the predicted spot prices is of greater magnitude. This relates to the discussion about the parameters we found, in Section 5.5. Furthermore using the expression for futures prices from Equation (5.3.19) and (5.3.20), and having the simulated spot prices and spread, a new dataset of futures prices can be calculated. This was also done 12 time steps ahead and for all maturities. Equation (5.3.19) enables us to obtain a dataset similar to the actual dataset from Bloomberg, meaning prices for 12 different futures contracts with the same maturity. This is due to holding the maturity T constant and keeping time t equal to 0, thus there is no discounting forward. However, equation (5.3.20) does discount forward by having t vary as time moves along. In Section 5.3.1, the relationship between futures prices, spread and the illiquid spot prices were shown by equation (5.3.2) and thus with this, the illiquid spot price can be calculated. All of these simulations and calculations represent the backbone for our further analysis, which will be presented in the next chapter.

90 83 5. The 3-factor Spot Price Model (a) Dated Brent (Spot Price) (b) ARA gasoil (Spot Price) Figure 5.4: Real vs. Simulated Spot Prices for Discussion What is good about our model? What are the benefits and concerns? These are essential questions to ask after having formulated a model that is used for estimating prices. Is the model realistic? Having read a large number of papers focused on this area of research, we have definitely tried to include the elements we saw realistic and important to incorporate. We wanted to take into account the long-term relationship between German gasoil and both Brent crude oil and ARA gasoil futures. This we did by modeling the log spread between these processes as stationary and mean reverting. We also wanted