Dynamic hedging strategies and commodity risk … hedging strategies and commodity risk management D. Lautier & A. Galli Mercantile Exchange, and American interest rates corresponding

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Dynamic hedging strategies and commodity risk management Final report

D. Lautier & A. Galli

Rapport pour l’Institut Français de l’Energie Januray 2005, Paris

Dynamic hedging strategies and commodity risk management D. Lautier & A. Galli

Introduction

This project prolongs the first report written in October 2002 for the Institut Français de l’Energie, which was untitled « Term structure models of commodity prices: elaboration and improvement ». This first report was centred on the analysis of the term structure of commodity prices, namely the relationship linking, at a specific date, all the futures prices on a derivatives market. It enhanced the understanding of the explicative factors of the prices curve and its dynamic behaviour. It also improved the representation of this curve with term structure models of commodity prices. Econometric methods adapted to these models (the Kalman filters) were developed and tested. Lastly, the performances of these models were appreciated and compared on the crude oil market.

This second project aims exploiting these results through the study of an application of term structure models: dynamic hedging strategies. This application relies on the information provided by futures prices. It can be exploited in the case of the crude oil market, but also in the case of natural gas or electricity.

Dynamic hedging strategies are used when:

- the maturity of the position in the physical market is longer than the maturity of the hedge instruments traded on the derivative market,

- the position hold in the physical market represents a huge volume. In that case, the hedging strategy requires the use of short term instruments because the liquidity of the derivatives market is concentrated on the nearest maturities.

Reflection on the use of term structure models for hedging purposes was motivated by Metallgesellchaft’s experiment. At the beginning of the 1990s, this firm tried to cover long-term forward commitments on the physical market with short-term futures contracts. This attempt ended in a resounding failure: USD 2.4 billion were lost. However, it initiated research, the goal of which was knowing whether such an operation could be safely undertaken. The interest of the Metallgesellschaft case is twofold. Firstly, being able to have long-term positions on the physical market while hedging the product of such sales constitutes an important stake. The success encountered by Metallgesellschaft when it proposed forward sales for a horizon of 5 or 10 years is indicative of that point. Secondly, Metallgesellschaft tried to hedge its long-term commitments in the physical market with short-term positions in the futures market. This kind

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of strategy is particularly interesting when the maturity of actively traded futures contracts is limited to a few months.

The theoretical framework of dynamic hedging strategies is the term structure models of commodity prices, because they rely on arbitrage reasoning and they naturally offer a means to undertake this kind of hedging strategies. This report will show how it is possible to compare the dynamic hedging strategies associated with different term structure models and, for each model, it will measure the efficiency of hedging strategies as a function of the choice of a specific maturity for the hedging position.

This report is divided into four parts. In the first one, we present the theoretical framework of the study, namely the term structure models of commodity prices. This review of the literature leads us, in a second part, to the selection of three specific models which will be used for dynamic hedging strategies. The third part presents the dynamic hedging strategies and the fourth one is devoted to the empirical results which were obtained on the crude oil market.

In the first part of the report, we review the literature on term structure models of commodity prices. We present the main models, from the simplest (one-factor models) to the more sophisticated versions (three-factor models) and we comment their theoretical backgrounds, as well as their respective drawbacks and advantages.

The analysis of the theoretical background is important because the term structure models are developed in a partial equilibrium framework. Thus, the choice of the nature of the factors (four different factors are generally used: the spot price, the convenience yield, the interest rates and the long-term equilibrium price) can be regarded as somehow arbitrary, because the state variables are supposed to be exogenous. However, a close examination of these models and a comparison with existing literature on price relationships in commodity markets show that these choices rely, on the one hand on the storage theory, and on the other hand on the study of the dynamic behaviour of the prices curve (Samuelson effect). Consequently, a model with exogenous spot price, convenience yield and/or long-term price can be regarded as the reduced form of a more general model in which these variables would be endogenously determined by production, consumption, and storage decisions. Moreover, two-factor models are probably the most interesting ones, because they allow for richer prices curves than one-factor models, and because until now, the interest of a third factor has not been proved.

The second part of the report presents the three term structure models chosen for the study of dynamic hedging strategies : a one-, a two-, and a three-factor model. Among these models, only the second one (the two-factor model) was used in the previous report written for the IFE.

The one- and two-factor models are those proposed by Schwartz [1997]. They are

currently well-known and extensively used. The single state variable of the first one is naturally

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the spot price. It is mean reverting, in order to ensure the realism of the model. The two-factor

model has been selected because it is an extension of the previous one. It is also mean reverting

and includes an additional variable: the convenience yield. The choice of the convenience yield

as a second state variable, rather than the long-term price, has been discussed in the first part of

the report. The three-factor model of Cortazar and Schwartz’ [2003] includes the spot price, the

convenience yield, and the long-term price. Thus, our choice insures continuity between the

different models, from the simplest to the most complicated. In this part of the report, we also

propose an analytical solution for each model.

The third part of the report is devoted to dynamic hedging strategies. More precisely, we show what conditions must be fulfilled in order to safely hedge a long term commitment on the physical market with short term positions in the futures market.

The hedging problem is separated into two steps. The first one relies on a naïve hedging strategy. Such strategies suggest taking a short (long) position in the futures market in order to hedge a long (short) position in the physical market, the two positions having the same size and expiration date. Thus, we consider that the delivery date of the commitment on the physical market and the expiry date of the futures contract are the same. This enables us to present the basic principles of traditional hedging. Naturally, traditional hedging cannot be initiated when the position to hedge has a horizon superior to the exchange-traded contracts. Thus, in the second step, we introduce a gap between the maturity of the positions on the physical and the futures markets. We show how the hedging positions are modified when dynamic hedging is undertaken.

Lastly, we present our analysis framework, which is inspired by Brennan and Crew (1997). Naturally, we worked with other models than the ones used by these authors. Our models are more sophisticated, and they have been proved to perform better. Moreover, our estimation methods are, by far, more precise.

In order to simplify the presentation, margin calls and deposits are ignored. We first recall of the commitment of the operator on the physical market. Then, we present the dynamic hedging strategies associated with the three term structure models that were retained, and we propose a way to determine their hedge ratios. Lastly, we expose the method that will be retained in order to study the efficiency of these dynamic hedging strategies, and we expose the concept of hedging error.

The fourth part of the report is devoted to empirical tests. First, we present the data, then the method used for the estimation of the models, and lastly the tests on the dynamic hedging strategies.

The data base is a powerful tool of our study because it is very complete. It is divided into two parts: crude oil futures prices corresponding to the crude oil contract of the New York

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Mercantile Exchange, and American interest rates corresponding to US government bonds. The data represent thus a large volume of information because we have all the curves, for crude oil prices as well as for interest rates: it represents roughly 50 000 futures prices, and 70 000 interest rates.

In order to test the dynamic hedging strategies associated with the term structure models, we needed the values of the parameters of the models. We thus applied the method that was explored in the previous report written for the IFE (namely Kalman filters) for the estimation of our three term structure models of commodity prices. The way to apply the Kalman filter to the two new models in exposed in the Appendix. We obtained the optimal parameters of the models, and we measured the relative performances of the models.

For the dynamic hedging strategies, we choose three different delivery dates for the commitment on the physical market. The maximum maturity is of 5 years, because the length of our database is also of 5 years. We selected the following maturities: 5 years, 3 years, and 18 months. We had thus the possibility to work with maturities that are longer than those used by Brennan and Crew. Indeed, the longer maturity, for these authors, was a 24 months maturity. We retained different sets of maturities for the hedging portfolio of each model, leading us finally to test and compare more than 1 200 different strategies on the crude oil market.

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Table of content

INTRODUCTION 2

PART 1. TERM STRUCTURE MODELS OF COMMODITY PRICES 10

1.1. Basic principles of contingent claim analysis 11

1.2. One-factor models 11

1.2.1. Spot price with a geometric Brownian motion 12

1.2.2 Mean reverting process 12

1.2.3. The assumptions concerning the convenience yield 13

1.3. Two-factor models 14

1.3.1. The convenience yield as the second state variable 14

The simple mean reverting process 15

The asymmetrical convenience yield 16

1.3.2. The long-term price as a second state variable 17

1.3.3. Convenience yield or long-term price ? 19

1.4. Three-factor models 21

1.5. Conclusion of the first part 23

PART 2. THREE TERM STRUCTURE MODELS OF COMMODITY PRICES 24

2.1. One factor model 24

2.2. Two-factor model 26

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2.3. Three-factor model 27

Appendix 2.1. Analytical solution of the one-factor model 29

1. Analytical solution 29

1.1. Mean 31

1.2. Variance 31

2. Volatility of futures returns 32

Appendix 2.2. Analytical solution of the two-factor model 33


1.1. Mean 35

1.2. Variance 36

Volatility of futures returns 39

Appendix 2.3. Analytical solution of the three-factor model 41


2. Volatility of futures returns 45

PART 3. DYNAMIC HEDGING STRATEGIES 46

3.1. Basic principles of traditional hedging 47

3.2. Dynamic hedging: presentation 48

3.2.1. The basic principles of dynamic hedging 48

3.2.2. Dynamic hedging strategies and term structure models 49

3.3. Dynamic hedging: the analysis framework 51

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3.3.1. The commitment 51

3.3.2. The hedging strategies associated with the three term structure models 51

3.3.3. The efficiency of the hedging strategies 52

3.3.3.1. The hedging error 53

3.3.3.2. Hedging error with transaction costs 54

Appendix 3. Hedge ratios associated with the three term structure models 55

1. The one-factor model 55

2. The two-factor model 56

3. The three-factor model 57

PART 4. EMPIRICAL TESTS 59

4.1. The data 59

4.2. Estimation of the models 61

4.2.1. The method of the Kalman filter 61

4.2.2. The optimal parameters 63

4.2.3. The performance criteria 64

4.2.4. The models performances 64

4.3. Dynamic hedging strategies 66

4.3.1. The strategies 66

4.3.2. The empirical results 67

4.3.2.1. The weights of the hedging portfolios 67

4.3.2.2. The efficiency of hedging strategies 78

4.3.2.3. Introducing the transaction costs : an example 81

4.4. Conclusion of the fourth part 82

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Appendix 4. Kalman filter and parameters’ estimation 84

1. Presentation 84

2. Parameters’ estimation 85

3. Application to the one-factor model 86

4. Application to the two-factor model 87

5. Application to the three-factor model 88

CONCLUSION OF THE REPORT 90

REFERENCES 91

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Part 1. Term structure models of commodity prices

This part of the report reviews the literature on term structure models of commodity prices. The term structure is defined as the relationship between the spot price and the futures prices for any delivery date. It provides useful information for hedging or investment decision, because it synthesizes the information available in the market and the operators’ expectations concerning the future. This information is very useful for management purposes: it can be used to hedge exposures on the physical market, to adjust the stocks level or the production rate. It can also be used to undertake arbitrage transactions, to evaluate derivatives instruments based on futures contracts, etc.

In many commodity markets, the concept of term structure becomes important, because the contracts’ maturity increases as the markets come to fruition. In the American crude oil market, this evolution has come a long way because since 1997, there are futures contracts for maturities as far as seven years. Thus, this market is today the most developed commodity futures market, considering the volume and the maturity of the transactions. It provides publicly available prices whereas in most commodities markets, the only information for far maturities is private and given by forward prices. The existence of these long term futures contracts authorizes empirical studies on the crude oil prices’ curves that were only possible before with forward prices. However, the informational content of the latter is not necessarily reliable or workable (forward contracts are not standardized, and the prices reporting mechanism does not force the operators to disclose their transactions prices).

Term structure models of commodity prices aim to reproduce the futures prices observed in the market as accurately as possible. They also provide a mean for the discovery of futures prices for horizons exceeding exchange-traded maturities. In this part of the report, we review the main term structure models of commodity prices, from the simplest (one-factor models) to the more sophisticated versions (three-factor models). Four different factors are generally used: the spot price, the convenience yield, the interest rate, and the long-term price.

Prior the presentation of the term structure models of commodity prices, because the latter borrow to the contingent claim analysis developed for options and interest rates models, a reminder of the basic principles of contingent claim analysis will be introduced.

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1.1. Basic principles of contingent claim analysis

The standard modelling procedure for pricing commodity derivatives typically follows the contingent claim analysis developed for options and interest rates. The term structure models of commodity prices share three assumptions: the market for assets is free of frictions, taxes, or transaction costs; trading takes place continuously; lending and borrowing rates are equal and they are no short sale constraints.

Then, the same method as the one developed in the context of interest rates is used to construct the model. First, the state variables (i.e. the uncertainty sources affecting the futures price) are selected and their dynamic is specified. Then, knowing that the price of a futures contract is a function of the state variables, the time, and the contract’s expiration date, using Itô’s lemma makes it possible to obtain the dynamic behaviour of the futures price. Afterwards, arbitrage reasoning and the elaboration of a hedge portfolio leads to the term premium and to the fundamental valuation equation characterizing the model. Finally, whenever it is possible, the solution of the model is obtained.

However, the transposition of the theoretical framework developed for interest rates in the case of commodities is not straightforward. The reasoning is indeed based on the assumption that the market is complete: in such a market, a derivative asset can be duplicated by a combination of other existing assets. If the latter are sufficiently traded to be arbitrage free evaluated, they can constitute a hedge portfolio whose behaviour replicates that of the derivative. Their proportions are fixed so that that there are no arbitrage opportunities and the strategy is risk-free. Thus, the return of the portfolio must be the risk free rate in equilibrium. Valuation is made in a risk-neutral world: it does not depend on the operators’ risk aversion.

The transposition problem arises from the fact that commodity markets are not complete. Markets of real assets are far from being free of arbitrage opportunities, as is the case for most financial markets. Thus, valuation will probably not be realised in a risk-neutral world for commodities markets, and several risk neutral probabilities may coexist.

1.2. One-factor models

A futures price is often defined as the expectation, conditionally to the available information at a date t, of the future spot price. Indeed, the spot price is the main determinant of the futures price. Thus, most one-factor models rely on the spot price1.

There have been several one-factor models in the literature on commodity prices. These

1 An exception is the one-factor model relying on the convenience yield developed by Veld-Merkoulova and de Roon [2003].

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models can be separated in step with the dynamic behaviour that is retained for the spot price: a geometric Brownian motion, or a mean reverting process. Moreover, the models can be distinguished in step with the assumption they retain concerning the convenience yield.

1.2.1. Spot price with a geometric Brownian motion

Brennan and Schwartz [1985], Gibson and Schwartz [1989] and [1990], Brennan [1991], Gabillon [1992] and [1995] use a geometric Brownian motion in their one-factor models. Among these models, Brennan and Schwartz’ model [1985] is the most well-known. It has been extensively used in subsequent research on commodity prices (see for example Schwartz [1998], Schwartz and Smith [2000], Nowman and Wang [2001], Cortazar, Schwartz and Casassus [2001], Veld-Merkoulova and de Roon [2003]).

The geometric Brownian motion is a dynamic commonly used to represent the behaviour of stock prices. When applied to commodities, the spot price’s dynamic is the following:

dztSdttStdS S )()()( σµ +=

where: - S is the spot price, - µ is the drift of the spot price, - is the spot price volatility, Sσ- dz is an increment to a standard Brownian motion associated with S.

The use of this representation implies that the variation of the spot price at t is supposed to be independent of the previous variations, and that the drift µ conducts the price’s evolution. The uncertainty affecting this evolution is proportional to the level of the spot price: when stocks are rare, S is high; in this situation, any change in the demand has an important impact on the spot price, because inventories are not sufficiently abundant to absorb the prices’ fluctuations.

Brennan and Schwartz’ model is probably the most simple term structure model of commodity prices and consequently it is very popular. However, the geometric Brownian motion is probably not the best way to represent the price dynamic. Indeed, the storage theory and the Samuelson effect show that the mean reverting process is probably more relevant.

1.2.2 Mean reverting process

Schwartz [1997], Cortazar and Schwartz [1997], Routledge, Seppi and Spatt [2000] retain a mean reverting process for their one-factor models. Among these models, that of Schwartz [1997], inspired by Ross [1995], is probably the most well-known2. In that case, the dynamic of the spot price is the following:

2 This model was also used by Schwartz and Smith [2000].

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( ) SS SdzdtSSdS σµκ +−= ln where: - S is the spot price, - κ is the speed of adjustment of the spot price, - µ is the long run mean log price, - σS is the spot price volatility - dzS is an increment to a standard Brownian motion associated with the spot price. In this situation, the spot price fluctuates around its long run mean. The presence of a speed of adjustment insures that the state variable will always return to its long-term mean µ. Therefore, two characteristics describe the spot price behavior. It has a propensity to return to its long-term mean, but simultaneously, random shocks can move it away from µ. The use of a mean reversion process for the spot price makes it possible to take into account the behaviour of the operators in the physical market. When the spot price is lower than its long run mean, the industrials, expecting a rise in the spot price, reconstitute their stocks, whereas the producers reduce their production rate. The increasing demand and the simultaneous reduction of supply have a rising influence on the spot price. Conversely, when the spot price is higher than its long run mean, industrials try to reduce their surplus stocks and producers increase their production rate, pushing thus the spot price to lower levels. This formulation of the spot price behaviour is preferable to the geometric Brownian motion, but it is not without flaws. For example, the storage theory shows that in commodity markets, the basis does not behave similarly in backwardation and in contango. Initially, the mean reverting process does not allow taking into account that characteristic.

The mean reverting process was also used by Cortazar and Schwartz in 1997 in a more sophisticated model. The authors introduce a variable convenience yield that depends on the deviation of the spot price to a long-term average price.

1.2.3. The assumptions concerning the convenience yield

Among the other one-factor models, the models studied by Brennan in 1991 are quite interesting because they all rely on a specific hypothesis concerning the convenience yield, which is an endogenous variable.

The first model was developed with Schwartz in 1985. In that case, the convenience yield is a simple linear function of the spot price S:

C(S) = cS

The second model expresses the convenience yield as a non-linear function of the price:

C(S) = a + bS + cS²

This formula is chosen because it is more flexible than the one used in 1985. However, it was afterwards forgotten, because the representation used in the third model, which is also non-

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linear, was preferred.

The advantage of the third model relies in the fact that it underlines that when the convenience yield is low, it cannot be lower than the opposite of the storage cost. The latter is supposed to be constant for a large spread of stocks levels, as long as the storage capacities are not saturated. The non-linearity comes from the fact that either the convenience yield is equal to the opposite of the storage cost, or it is proportional to the spot price:

C(S) = max (a, b + cS)

This study of the convenience yield is very interesting because it is the support of further researches. Indeed, with these models, Brennan uses for the first time a non-linear representation of the behaviour of the convenience yield, which has been quite intensively exploited later. However, Brennan’s empirical study [1991], as well as Gibson and Schwartz’ article [1990], lead to the conclusion that there are limits to one-factor models. These models are indeed attractive because they are very simple, but they provide a very summary representation of the prices curve, except for very short maturities and for some commodity like gold. This limit of the one-factor models is not specific of commodity markets. The same critique can be invoked in the case of interest rates.

1.3. Two-factor models

The homogeneity in the choice of the state variables disappears when a second stochastic variable is introduced in term structure models of commodity prices. Most of the time, the second state variable is the convenience yield. However, models based on long-term price have also been developed. In all such models, the introduction of a second state variable allows for richer shapes of curves than one-factor models (especially for long term maturities) and richer volatility structures. Yet this improvement is rather costly, because two-factor models are more complex.

1.3.1. The convenience yield as the second state variable

When the convenience yield is considered as an uncertainty source affecting the futures price, it is supposed – contrary to what was done in one-factor models – that the convenience yield provides an information which is not totally included in the spot price. In other words, the convenience yield is not perfectly correlated to the spot price. This imperfect correlation can be explained in the following way: in a commodity market, a change in the transformation or transportation rate can have an impact on the implicit revenue associated with stock holding, and simultaneously no influence on the spot price, because this change does not lead to a transaction in the physical market.

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The simple mean reverting process

Schwartz’ model [1997] is probably the most famous term structure model of commodity prices. It was used as a reference to develop several models that are more sophisticated (Hilliard and Reis [1998], Schwartz [1998], Neuberger [1999], Schwartz and Smith [2000], Lautier and Galli [2001], Yan [2002], Richter and Sorensen [2002], Veld-Merkoulova and de Roon [2003], Cortazar and Schwartz [2003]).

Inspired by the one proposed by Gibson and Schwartz in 1990, the latest model is more tractable than its former version: it has an analytical solution. The two-factor model supposes that the spot price S and the convenience yield C can explain the behavior of the futures price F. The dynamic of these state variables is:

( )[ ]

+−=+−=

CC

SS

dzdtCkdCSdzSdtCdS

σασµ )(

with: κ, σS, σC >0 where: - µ is the drift of the spot price,

- Sσ is the spot price volatility, - dzS is an increment to a standard Brownian motion associated with S, - α is the long run mean of the convenience yield, - κ is the speed of adjustment of the convenience yield, - is the volatility of the convenience yield, Cσ- dzC is an increment to a standard Brownian motion associated with C.

The idea of a mean reverting process is retained in this model. However, it is applied to the convenience yield. This choice is not incoherent with what was done in the case of one-factor models, because the convenience yield intervenes in the spot price dynamic and gives it – though marginally – a mean reverting tendency. The convenience yield becomes then a stochastic dividend affecting the spot price dynamic. Such a formulation is a good illustration of the fact that the convenience yield is an implicit revenue associated with physical stocks. Moreover, it authorizes some comparisons with other financial assets such as bonds and securities.

When applied to the convenience yield, the Ornstein-Uhlenbeck process relies on the hypothesis that there is a regeneration property of inventories, namely that there is a level of stocks, which satisfies the needs of industry under normal conditions. Once again, the behaviour of the operators in the physical market guarantees the bearing of this normal level. When the convenience yield is low, the stocks are abundant and the operators sustain a high storage cost compared with the benefits related to holding the raw materials. Therefore, if they are rational, they try to reduce these surplus stocks. Conversely, when the stocks are rare the operators tend to reconstitute them.

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Moreover, as the storage theory shows, the two state variables are correlated. Both the spot price and the convenience yield are indeed an inverse function of the inventory level. Nevertheless, as Gibson and Schwartz [1990] have demonstrated, the correlation between these two variables is not perfect. Therefore, the increments to standard Brownian motions are correlated, with:

[ ] dtdzdzE CS ρ=×

where ρ is the correlation coefficient.

This model, which is quite tractable, presents a limit. Indeed, it ignores that in commodity markets, the price’s volatility is positively correlated with the degree of backwardation. This phenomenon has been widely reported and commented (see for example Williams and Wright [1991], Ng and Pirrong [1994], Litzenberg and Rabinowitz [1995]) and can be explained by the examination of arbitrage relationships between the physical and the futures markets. Such a study shows that the basis has an asymmetrical behavior: in contango, its level is limited to storage costs. This is not the case in backwardation:

“Arbitrage can always be relied upon to prevent the forward price from exceeding the spot price by more than net carrying cost… [but] can not be equally effective in preventing the forward price from exceeding the spot price by less than net carrying cost.” (Blau, [1944]).

Furthermore, the basis is stable in contango, and volatile in backwardation, since in the latter situation stocks cannot absorb price fluctuations.

The asymmetrical convenience yield

This behavior of the basis in commodity markets can lead to the assumption that the convenience yield is an option (Heinkel, Howe and Hughes [1990], Milonas and Tomadakis [1997], Milonas and Henker [2001]) or that it has an asymmetrical behavior. This assumption has been introduced in term structure models by Brennan [1991], Routledge, Seppi, and Spatt [2000], and Lautier and Galli [2001].

Brennan [1991] introduces an asymmetric convenience yield in his model because he takes into account a non negativity constraint on inventory. However, he supposes that the convenience yield is deterministic. In the model presented by Routledge et alii, the asymmetry in the behavior of the convenience yield is introduced in the correlation between the spot price and the convenience yield. This correlation is higher in backwardation than in contango. The convenience yield is endogenous, and it is determined by the storage process. However, it is stochastic. The two factors are the spot price and exogenous transitory shocks affecting supply and demand. Lautier and Galli [2001] propose a two-factor model inspired by Schwartz’ model [1997], in which the convenience yield is also mean reverting and acts as a continuous dividend.

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An asymmetry is however introduced in the convenience yield dynamic: it is high and volatile in backwardation, when inventories are rare. It is conversely low and stable when inventories are abundant. The asymmetry is measured by the parameter β. When the latter is set to zero, the asymmetrical model is reduced to Schwartz’ model.

1.3.2. The long-term price as a second state variable

Another approach of the term structure of commodity prices consists in considering the decreasing pattern of volatilities along the prices curve. In that situation, it is possible to infer that the two state variables correspond to the two extremities of the prices curve, the spot and the long-term price. This kind of approach was followed by Gabillon [1992] and Schwartz and Smith [2000].

Gabillon [1992] uses the spot and long-term prices as state variables. In this model, the convenience yield is an endogenous variable, dependent on the two factors. The use of the long-term price as a second state variable is justified by the fact that this price can be influenced by elements that are exogenous to the physical market, such as expected inflation, interest rates, or prices for renewable energies. Thus, the spot and long-term prices reassemble all the factors allowing the description of the term structure movements. The author retains a geometric Brownian motion to represent the behaviour of the long-term price. Moreover, the two state variables are assumed to be positively correlated.

Schwartz and Smith [2000] propose a two-factor model, where the state variables come from the decomposition of the spot price. This decomposition authorizes the distinction between short-term variations and long-term equilibrium:

tttS ξχ +=)ln(

where : - St is the spot price at t, - tχ is the short-term deviation in prices, - tξ is the equilibrium price level.

These two factors are not directly observable, but they are estimated from spot and futures prices. More precisely, the movements in prices for long-maturity futures contracts provide information about the equilibrium price level, whereas the differences between the short and long-term futures prices provide information about short-term variations in prices. This model does not explicitly consider changes in convenience yields over time, but it is equivalent to the two-factor model proposed by Gibson and Schwartz in 1990, in that the state variables in one of the models can be expressed as linear combinations of the state variables in the other model.

The short-term deviation is assumed to revert to zero, following an Ornstein-Uhlenbeck process. The equilibrium level is assumed to follow a Brownian motion process. Thus, the

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dynamic of these two state variables is the following:

+=

+−=

ξξ

χχ

σµξ

σκχχ

dzdtddzdtd

t

tt

where: - κ is the speed of adjustment of the short-term deviation, - χσ is the volatility of the short-term prices, - is an increment to a standard Brownian motion associated with χdz tχ , - µ is the drift of the equilibrium price level, - is the volatility of the equilibrium price level, ξσ- is an increment to a standard Brownian motion associated with ξdz tξ

Changes in the short-term deviations represent temporary changes in prices (e.g. caused by abrupt weather alteration, supply disruption, etc) and are not expected to persist. They are tempered by the ability of market participants to adjust to inventory levels in response to changing market conditions. Changes in the long-term level represent fundamental modifications, which are expected to persist. The latter are due, for example, to shifts in the number of producers in the industry. The long-term equilibrium is also determined by expectations of exhausting supply, improving technology for the production and discovery of the commodity, inflation, as well as political and regulatory effects.

The most important advantage of these models is that they avoid the questions concerning the convenience yield, its estimation, and its economic significance (on this particular point, see for example Williams and Wright [1989], Brennan, Williams and Wright [1997], Frechette and Fackler [1999]). The idea of a long-term equilibrium is also in line with recent works on long memory3 in the commodity futures markets. Long memory in convenience yields has been studied by Mazaheri [1999], and Barkoulas, Labys and Onochie [1999] showed that there is long memory in futures prices.

However, using the long-term price as a state variable gives rise to two critiques. First, nothing is said about the horizon of the long-term equilibrium. Second, these models regard as stochastic a variable which is a priori stable. Suppose for example that the long term price of the crude oil can be approached by futures contracts having a delivery date of seven years. The average trading volume of these contracts, since their launch in 1997, is 35 per day… Several weeks can be spent without a single transaction. As a comparison, the daily trading volume on the nearest futures contract, since 1997, represents 63 153 contracts.

3 Long memory or long-term dependence describes the correlation structure of series at long lags. If a series exhibits long memory, there is persistent temporal dependence between distant observations. Such series are characterized by distinct but non-periodic cyclical patterns.

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1.3.3. Convenience yield or long-term price ?

In a two-factor model, a choice must be made when defining the second state variable: either the convenience yield or the long-term price can indeed be chosen. Until now however, even if a strong preference has been granted to the convenience yield, there is no quantifiable element justifying the choice of one or the other variables. Indeed, no comparison was made between the different models. Thus, whereas it is obvious that each factor provides an important information, it is impossible to know whether one brings more than the other.

The choice of the long-term price is interesting because it avoids the questions concerning the convenience yield and its economic significance. Indeed, even if this concept dates from 1939, it was sometimes questioned. It seems however that these critiques are not really founded, for the following reasons.

The difficulties associated with the concept of convenience yield come first of all from the time necessary to define it. Several authors indeed have followed one another in order to propose, by turns, an incomplete definition. Kaldor (1939) was mainly interested in the supply function of inventories and he saw in the convenience yield a way to respond to the costs associated with the repetition and the waiting of supply. Working (1949) looked into the characteristics of the transformation of raw materials and he considered mainly the fact that these activities sustain high fixed costs. In this context, inventories appear as a means to maintain the continuity of exploitation and to cover a part of the fixed costs. Lastly, Brennan (1958) studied the demand of commodity and he mostly regarded inventories as a way to profit from an unexpected and favourable change in the demand.

Several decades have passed before these definitions are reassembled into a coherent one, by Brennan and Schwartz in 1985. Indeed, since that date, all authors refer themselves to this text:

“The convenience yield is the flow of services that accrues to an owner of the physical commodity but not to the owner of a contract for future delivery of the commodity. [...] Recognizing the time lost and the costs incurred in transporting a commodity from one location to another, the convenience yield may be thought of as the value of being able to profit from temporary local shortages of the commodity through ownership of the physical commodity. The profit may arise either from local price variations or from the ability to maintain a production process as a result of ownership of an inventory of raw material”.

Beyond these problems of definition, the convenience yield was criticized for the difficulties associated with its estimation. Two kind of objection have been raised. The first, initiated by Williams and Wright (1989), asserts that the convenience yield, when it is estimated with data on inventories, can be overestimated. Indeed, inventories sometimes regroup

Cerna 19


commodity which are not real substitutes, because of quality differentials and/or transportation costs. This objection is totally justified. It could have lead to a real question on the existence of the convenience yield. However, this existence was later firmly established, even if this eventual bias is taken into account, by Brennan, Williams and Wright (1997) and by Frechette and Facker (1999). The second category of objection relies on the non-observable nature of the convenience yield. Indeed, even if it is possible to proxy the value of the convenience yield with data on inventories or futures prices, there is no real traded asset corresponding to this variable. This characteristic undoubtedly makes tricky the estimation of the convenience yield. It does not signify, however, that the objection is receivable. All expected variables, in the field of finance, are not directly observable. Does it question, for example, the notion of expected inflation?

It is also possible to note that, when applied to bonds, currencies, securities or stock indexes, the term structure involves several common factors: the risk-free interest rate, the risk premium, and the yield associated to the possession of the underlying asset. There is obviously no reason for this last factor to disappear in the case of commodities.

Thus, it seems not relevant to choose the long-term price in order to avoid some difficulties associated with the convenience yield. Moreover, certain difficulties must also be undertaken with the long-term price: it is also not observable and it is rather stable. Yet, this latter characteristic is not really valuable for a stochastic variable.

Maybe the choice could be guided by the use that one wish to have of a term structure model of commodity prices. It seems indeed, a priori, that the convenience yield and the storage theory lead to short or medium term applications, whereas the introduction of a long term price leads to a long-term analysis.

To give an exhaustive view of the two-factor models, one must quote the researches conducted on the seasonality of commodity prices. In this field, Gabillon once again forged new territory in 1992 with a model including a seasonal function as a composite of sine and cosine functions. The same type of formalization was retained by Richter and Sorensen [2002]. However, the latter model takes the spot price and the convenience yield as state variables, whereas Gabillon retains the spot and the long-term prices.

The most important advantage of the two-factor models is that, contrary to the one-factor models, they authorize a good representation of the prices curves.

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1.4. Three-factor models

Until 1997, every term structure model of commodity prices assumed the interest rate constant. Such a hypothesis amounts to saying that the term structure of interest rates is flat, which is all the more reductive as the horizon of analysis is remote. Schwartz, in 1997, proposes a model including three state variables: the spot price, the convenience yield and the interest rate. The latter has a mean reverting behaviour. More precisely, the dynamic of the state variables is the following:

( )( )

( )

+−=+−=

+−=

rr

CC

SS

dzdtrmadrdzdtCdC

dzdtCrSdS

σσακ

σ

where: - µ is the drift of the spot price, - α is the long run mean of the convenience yield, - κ is the speed of adjustment of the convenience yield, - a is the speed of adjustment of the interest rate, - m is the long run mean of the interest rate, - is the volatility of the variable i, iσ- dzi is an increment to a standard Brownian motion associated with the variable i.

The introduction of a stochastic interest rate in the analysis of prices relationship is important from a theoretical point of view: the assumption of a constant interest rate amounts to saying that futures and forward prices are equivalent, which is not the case (Cox, Ingersoll, Ross [1981]). With a stochastic interest rate, it is possible to take into account the margin call mechanism of the futures market. Thus, two distinct pay-off structures can be taken into account for futures and forward contracts. Finally, the presence of the interest rate as a third explicative factor of the futures price is consistent with the storage theory.

Since 1997, several three-factor models were proposed. In 1998, Hilliard and Reis modified the three models proposed by Schwartz in 1997. They introduced jumps in the spot price process, in order to take into account the large and abrupt changes, due to supply and demand shocks affecting certain commodity markets, especially the energy commodities used for heating. In 2000, Schwartz and Smith proposed an extension of their short-term/long-term model in which the growth rate for the equilibrium price level is stochastic. Such an extension improves the model’s ability to fit long-term futures prices. Another improvement of three factors models was proposed by Yan in 2002. In his model, he incorporates stochastic convenience yields, interest rates and volatility. He also introduces simultaneous jumps in the spot price and volatility. The convenience yield follows an Ornstein-Uhlenbeck process, whereas the interest rate follows a square-root process, and the volatility follows a square-root jump-diffusion process. However, Yan finds that stochastic volatility and jumps do not alter the futures price at a given point in time. Nevertheless, they play important roles in pricing options

Cerna 21


on futures.

Lastly, Cortazar and Schwartz [2003] proposed a three-factor model related to Schwartz [1997], in which all three factors are calibrated using only commodity prices4. In this model, the authors consider as a third risk factor the long-term spot price return, allowing it to be stochastic and to return to a long-term average. The two other stochastic variables are the spot price and the convenience yield. The convenience yield models temporary variations in prices due to changes in inventories, whereas the long-term return is due to changes in technologies, inflations or demand pattern. The dynamic of the state variable is the following:

( )

( )

+−=+−=

+−=

33

22

11

dzdtvvadvdzydtdy

SdzSdtyvdS

σσκ

σ

where:

- S is the spot price - y is the demeaned convenience yield, with y = C - α, where α is the long run mean

of the convenience yield C - v is the expected long-term spot price return; with v = µ - α, where µ is the drift of

S - κ is the speed of adjustment of the demeaned convenience yield - a is the speed of adjustment of v - v is the long run mean of the expected long-term spot price return - σi is the volatility of the variable i - dzi is the increment of a standard Brownian motion associated with the variable i

Until now, the interest of a third factor in a term structure model of commodity prices has not been proved. As far as the interest rate is concerned, the main argument in favour of its introduction is theoretical. It consists in asserting that, for long-term analysis, considering that the term structure of interest rates is flat is awkward. Consequently, it seems relevant to integrate a stochastic factor corresponding directly or not, to the interest rate. However, this argument is weak in practice. First, the principal component analysis of the crude oil prices curve (Lautier and Galli, 2002) has shown that even for maturities reaching seven years, the part of the third factor in the total variability of futures prices is marginal. Second, empirical tests carried out with Schwartz’ two-factor model show that its performances can be excellent, even for long-term futures prices. Third, Schwartz himself, when comparing his two- and three-factor models concluded that the two models are empirically similar. Lastly, when Cortazar and Schwartz propose a three-factor model in 2003, where the third factor is the long-term price, they underline that such a model is useful only for certain specific cases empirically observed. Thus, the improvement of realism obtained by introducing a third factor seems weak compared

4 In 1997, Schwartz calibrated the third factor of his model (the interest rate) using bond prices.

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with the rise in the complexity of the analysis. Moreover, there is a need for parsimony when the models are conceived for the evaluation of more complex derivatives products.

1.5. Conclusion of the first part

The term structure models presented in this part of the report are developed in a partial equilibrium framework. Thus the choice of the nature of the factors can be regarded as somehow arbitrary, because the state variables are supposed to be exogenous. However, a close examination of these models and a comparison with existing literature on price relationships in commodity markets show that these choices rely, on the one hand on the storage theory, and on the other hand on the study of the dynamic behaviour of the prices curve (Samuelson effect). Consequently, a model with exogenous spot price, convenience yield and/or long-term price can be regarded as the reduced form of a more general model in which these variables would be endogenously determined by production, consumption, and storage decisions.

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Part 2. Three term structure models of commodity prices

This part of the report exposes the three term structure models chosen for the study of

dynamic hedging strategies: a one-, a two-, and a three-factor model.

The one- and two-factor models are those proposed by Schwartz [1997]. They are

currently well-known and extensively used. The single state variable of the first one is naturally

the spot price. It is mean reverting, in order to ensure the realism of the model. This two-factor

model has been selected because it is an extension of the previous one. It is also mean reverting

and includes an additional variable: the convenience yield. The choice of the convenience yield

as a second state variable, rather than the long-term price, has been discussed in the first part of

the report. Moreover, the two-factor model of Schwartz [1997] and the two-factor model and

Schwartz and Smith [2000] can be considered as two different versions of the same model. The

three-factor model of Cortazar and Schwartz’ [2003] includes the spot price, the convenience

yield, and the long-term price. This model also insures continuity between the previous ones.

The Appendix provides the analytical solutions of these models. It also presents the

analysis of the volatility of the futures prices returns

2.1. One factor model

In the one-factor model, the single state variable is the spot price, which dynamic is the following:

( ) SdzSdtSdS σµκ +−= ln

with : κ, σ >0

where: - S is the spot price,

- µ the long-run mean of lnS,

- κ is the speed of adjustment of lnS,

- σ is the volatility of the spot price,

- dz is the increment of a standard Brownian motion associated with S.

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Defining X = ln S and applying Ito’s Lemma, this implies that the log price can be

characterized by an Ornstein-Uhlenbeck stochastic process:

( ) dzdtXdX σακ +−=

with : κ

σµα2

2

−=

An arbitrage reasoning and the construction of a hedging portfolio leads to the fundamental valuation equation of the futures prices, which is:

( ) 0ln21 22 =−−+ τακσ FSFSF SSSS

with : κλµα −=

where λ is the risk premium associated with the state variable.

The terminal boundary condition associated with this equation is:

F(S,T,T) = S(T)

It represents the convergence of the futures and the spot prices at the contract’s expiration. This convergence is due to the possibility to deliver the commodity at maturity. It is insured by arbitrage operations between the physical and the paper markets.

The solution of the model5 expresses the relationship at t between an observable futures price F for delivery in T and the state variable S:

( ) ( )

−+−+= −− tt eeSTtSF κκ

κσαα 2

2

14

lnˆexp),,(

The term structure of proportional futures volatility, in this model, is given by

tF e κσσ 222 −=

Thus, a feature of this model is that as the time to maturity of the futures contract

approaches infinity, the volatility of its price converges to zero. Meanwhile, the futures price

converges toward a fixed value, which is independent of the spot price:

( )

+=∞

κσα4

exp,2

SF

5 The solution of the models can be obtained with a Feynman-Kac solution.

Cerna 25


In this model, when the spot price is above this infinite maturity futures price, the term

structure will be in backwardation. Conversely, when the spot price is below it will be in

contango. This is the main limit of this model, which does not allow for much flexibility in the

prices curves.

2.2. Two-factor model

The two-factor model supposes that the spot price S and the convenience yield C can explain the behavior of the futures price F. The dynamic of these state variables is:

( )[ ]

+−=+−=

CC

SS

dzdtCkdCSdzSdtCdS

σασµ )(

with: κ, σS, σC >0 where: - µ is the drift of the spot price,

- Sσ is the spot price volatility, - dzS is an increment to a standard Brownian motion associated with S, - α is the long run mean of the convenience yield, - κ is the speed of adjustment of the convenience yield, - is the volatility of the convenience yield, Cσ- dzC is an increment to a standard Brownian motion associated with C.

The increments to standard Brownian motions are correlated, with:

[ ] dtdzdzE CS ρ=×

where ρ is the correlation coefficient.

An arbitrage reasoning and the construction of a hedging portfolio leads to the solution of the model. It expresses the relationship at t between an observable futures price F for delivery in T and the state variables S and C:

( )

+

−−×=

−

τκ

κτ

BetCtSTtCSF 1)(exp)(),,,(

with: ( )

−×

−++

−×+

×

−+−=

−−

2

2

3

22

2

2 1142 κκ

σρσσκακ

στκ

ρσσκ

σατκτκτ eerB C

CSCCSC

- ( )κλαα /−=

where: - r is the risk free interest rate, assumed constant, - λ is the market price of convenience yield risk, - τ = T - t is the maturity of the futures contract.

Cerna 26


With the two-factor model, we can obtain various shapes of prices curves: they can be

sunken, humped, or flat. The level of the futures prices is a decreasing function of the

convenience yield: the more the latter increases, the more the price level diminishes. Moreover,

the gap between the convenience yield and its long run mean α influences the shape of the

prices curves. When the convenience yield is far from its long run value, it takes a long time

until the curve becomes stable.

Thus, the introduction of a second state variable makes it possible to obtain richer prices

curves than with a one-factor model (this remark is also valuable in the field of interest rates).

Furthermore, Schwartz’ model is also more realistic that the one of Brennan and Schwartz,

because in the two-factor model the volatility of the futures prices decreases with the maturity τ:

×

−×−

−+=

−−

CSCSFee σρσκκ

σστσκτκτ 121)(

2222

When the contract reaches its expiration date, the futures price’s volatility converges towards

the spot price’s volatility. Conversely, when the maturity tends towards infinity, the volatility of

the futures price tends towards a fixed value:

κσρσ

κσ

σστ

CSCSF

2lim 2

222 −+=

∞→

However, the two-factor model it is also more complex, because it includes six

parameters, as opposed to two for the one-factor model.

2.3. Three-factor model

In this model, the three state variables are the spot price, the convenience yield, and the long-term spot price return. The dynamic of these state variables is the following:

( )

( )

+−=+−=

+−=

33

22

11

dzdtvvadvdzydtdy

SdzSdtyvdS

σσκ

σ

with:

dtdzdz 1221 ρ= dtdzdz 1331 ρ= dtdzdz 2332 ρ=

Cerna 27


where: - S is the spot price - y is the demeaned convenience yield, with y = C - α, where α is the long run mean

of the convenience yield C - v is the expected long-term spot price return; with v = µ - α. µ is the drift of S - κ is the speed of adjustment of the demeaned convenience yield - a is the speed of adjustment of v - v is the long run mean of the expected long-term spot price return - σI is the volatility of the variable i - ρij is the correlation between the variable I and j - dzi is the increment of a standard Brownian motion associated with the variable i

The fundamental valuation equation of the model is:

( ) ( ) ( )[ ] 021

21

21

321

23321331122123

22

221

=−−−+−−+−−+

+++++

τλλκλ

ρσσρσσρσσσσσ

FFvvaFySFyv

FFSFFFFS

vyS

yvSvSyvvyySS

The solution of this equation is:

( )

( )

( ) ( )

( ) ( ) ( )

+−−

−−

++

++

+−+−−−

−++−

+−−+−+

−+−

+−−

+−

−

=

+−

−−

−−

−−

−−−

−−−

τκκκκ

κτκκκ

κκρσσ

τσ

τρσσλ

κτκ

σ

κτκ

ρσσλτλ

κ

τ

τκ

κτκτ

ττ

ττ

τκτκτ

κττκτ

aaaae

eaaeaaee

aaaee

a

eaa

vaee

eaevey

SvySF

a

aa

aa

a

a

22

2

2

22

2223322

3

23

2133132

3

22

212212

1

3244

13244

111

exp,,,

The volatility of futures returns is the following:

( ) ( ) ( ) ( ) ( ) ( )( )κ

ρσσρσσκ

ρσσσκ

σστστκττκττκτ

aee

aee

aee aaa

F

−−−−−− −−−

−+

−−

−+

−+=

1121212112332133112212

2232

222

21

2

which converges, as τ goes to infinity, to:

( )κ

ρσσρσσκ

ρσσσκσστσ

aaaF233213311221

2

23

2

222

12 222

−+−++=∞→

This volatility term structure decreases with maturity and converges to a positive constant. This is consistent with a mean reverting non stationary process.

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Appendix 2.1. Analytical solution of the one-factor model

The first part of this appendix is devoted to the analytical solution of the two-factor

model. The second part presents the analysis of the volatility of the futures prices returns.

1. Analytical solution

The partial derivatives equation to solve is the following:

( ) 0ln21 22 =−−+ τακσ FSFSF SSSS

with : κλµα −=

and : F(S, C, T, T) = S(T)

where λ is the risk premium associated with the state variable.

In order to solve this equation, we use a Feynman-Kac solution. In a risk-neutral world, it is

possible to consider the future prices F(S, t, T) as the expectation, at t and under the probability

Q, of the spot price at T:

)]([),,( TSETtSF Qt=

Originally, the dynamics of the state variable is the following:

( ) SdzSdtSdS σµκ +−= ln

with : κ, σ >0

where : - S is the spot price,

- µ the long-run mean of lnS,

- κ is the speed of adjustment of lnS,

- σ is the volatility of the spot price,

- dz is the increment of a standard Brownian motion associated with S,

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Letting X = ln S, we have:

2)(21 dSXdSXdtXdX SSSt ++=

( )( ) ( )dtSXSdzSdtSXdX SSS22

21ln σσµκ ++−=

( ) dtdzdtSdX 2

21ln σσµκ −+−=


with : κ

σµα2

2

−=

In the absence of arbitrage opportunity, this dynamic, under the risk-neutral probability

Q, is the following:

( ) *ˆ dzdtXdX σακ +−=

with : κλ

κσµα −−=2

2

The dynamic of X can be found with a change in the variable X.

Let us define: ( ) tetXt καY −= )()(

...++= dXYdtYdY Xt

...++= dXeYdtdY ktκ

( ) ( ) dzedttXedtetXdY ktktt σακακ κ +−+−= )()(

dzedY ktσ=

)()0()(0

sdzeYtYt

ks∫+= σ

( ) )()0()(0

sdzeeXtYt

kst ∫+−= σα κ

Cerna 30


[ ] ∫−− +−+=t

OC

sC

tt sdzeeeXtX )()0()( *κκκ σαα

The Feynman-Kac solution necessitates the calculation of the mean and variance of

X(t).

1.1. Mean

[ ] [ ] teXtXE καα −−+= )0()(

1.2. Variance

[ ] [ ] [ ]( )22)( XEXEtXVar −=

● E[X²]

[ ] ( ) ( )

+−+−+= ∫−−−

2

0

2222 )()0(ˆ2)0()(t

sttt sdzeeEeXeXtXE κλκκ σαααα

( )tt

st esdzeeE κκλ

κσσ 2

22

0

12

)( −− −=

∫

● E[X]²

[ ] ( ) ( ) tt eXeXtXE κκ αααα −− −+−+= )0(ˆ2)0()( 2222

⇒ [ ] ( )tetX κ

κσ 2

2

12

)( −−=Var

Thus, with: )](var[

21)]([

)]([),,(TXTXEQ

t

QteTSETtS

+==F

( ) ( )

−+−+= −− tt eeSTtSF κκ

κσαα 2

2

14

lnˆexp),,(

Cerna 31


2. Volatility of futures returns

++= 2)(

21)(1 dSFdSFdtF

FFdF

SSSt

tF e κσσ 222 −=

( )

+=∞

κσα4

exp,2

SF

Cerna 32


Appendix 2.2. Analytical solution of the two-factor model

The first part of this appendix is devoted to the analytical solution of the two-factor




( ) ( )[ ] OFCFCrSFSFFSF CSSCCSCCCSSS =−−−+−+++ τλακσρσσσ 222

21

21

with : F(S, C, T, T) = S(T)

In order to solve this equation, we use a Feynman-Kac solution. In a risk-neutral world,

it is possible to consider the future prices F(S, C, t, T) as the expectation, at t and under the

probability Q, of the spot price at T:

)]([),,,( TSETtCSF Qt=

Originally, the dynamics of the state variables is the following:

( )( )

+−=+−=

CC

SS

dzdtCdCSdzSdtCdS

σακσµ

with : [ ] dtdzdzE CS ρ=×

κ, σS, σC >0

where : - S is the spot price,

- C is the convenience yield associated with inventories,

- µ is the drift,

- α is the long-run mean of C,

- κ is the speed of adjustment of C toward α,

- is the volatility of the spot price, Sσ

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- is the volatility of the convenience yield, Cσ

- dzS is the increment of a standard Brownian motion associated with S,

- dzC is the increment of a standard Brownian motion associated with C,

- ρ is the coefficient correlation between the two Brownian motions.

In the absence of arbitrage opportunity, this dynamic, under the risk-neutral probability

Q, is the following:

( )[ ]

+−−=

+−=

)(*

*

CC

SS

dzdtCkdC

SdzSdtCrdS

σλα

σ

with : [ ] dtdzdzE CS ρ=× **

where : - r is the risk free interest rate, assumed constant,

- z and z are the standard Brownian motions respectively associated with S and C

under the probability Q.

*S

*C

Letting G = ln S, and applying the Itô’s lemma to this relation, we obtain the dynamic of

G:

[ ] [ ]

21

21)(

²/1 /1 :with

)(21

*2

22*

2

sSS

SSSsSS

SS

S

SSSt

dzdtCrdG

dtSGSdzdtCrSGdG

SGSG

dSGdSGdtGdG

σσ

σσ

+

−−=⇔

++−=⇒

−==

++=

⇒ G T G t T t r v dv C v dv dz yS St

T

t

T

t

T

( ) ( ) . ( ) ( ) ( ) ( )*= − − + − + ∫∫∫05 2σ σ s

In this model, interest rates are assumed constant. Thus: r(v) = r.

Letting : (T-t) = τ :

)()(5.0)( *2 ydzdvvCrtGG(T)T

t

T

tsSS ∫ ∫+−+−= σττσ

The Feynman-Kac solution leads to compute the mean and variances of G(T).

Cerna 34


1.1. Mean

[ ] [ ]E G(T) G t r E C v dvtQ

S tQ

t

T

= − + − ∫( ) . ( )05 2τσ τ

In order to solve this equation, we must know E[C(v)], in a risk neutral framework. We

have the following dynamic:

( )[ ]C dt dzC C= − − +κ α λ σ *dC

Letting α the long run mean of the convenience yield, netted by the risk premium:

a = −αλκ

⇒ )( *CCdzdtCdC σακ +−=

Letting : Y ( ) tetCt κα−= )()(

...++= dCYdtYdY Ct

...++= dCeYdtdY ktκ

( ) ( ) dzedttCedtetCdY ktktt σακακ κ +−+−= )()(

dzedY ktσ=

)()0()(0

sdzeYtYt

ks∫+= σ

( ) )()0()(0

sdzeeCtYt

kst ∫+−= σα κ

[ ] ∫−− +−+=t

OC

sC

tt sdzeeeOCtC )()()( *κκκ σαα

⇒ [ ] ∫−−− +−+=v

tC

sC

vtv sdzeeetCvC )()()( *)( κκκ σαα

⇒ )(])([)]([ tvQt etCvCE −−−+= καα

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⇒ We introduce this expression in this equation:

[ ] [ ]∫−−+=T

ttS

Qt dvvCErtGG(T)E )(5.0)( 2τστ

⇒ [ ] ( )dvetCrtGG(T)ET

t

tvS

Qt ∫ −−−+−−+= )(2 ))((5.0)( καατστ

⇒ [ ]

−−−−+= ∫ −− dvetCrtGG(T)E

T

t

tvS

Qt

)(2 ))((5.0)( κατατστ

Letting: κ

κτ−−=

eH 1 , we have:

[ ] ]))([(5.0)( 2 HtCrtGG(T)E SQt ×−−−−+= ατατστ

1.2. Variance

−+

= ∫ ∫∫

T

t

T

tSSS

T

t

ydzdvvCCovdvvCVarTGVar )(,)(2)()]([ *2 στσ

with Var[dz]= dt = τ

⇒

−+

= ∫ ∫∫∫ ∫ −−

T

t

T

tSS

v

tC

svCS

T

t

v

tC

svC ydzsdzedveCovsdzedveVarTGVar )(,)(2)()]([ **2* σστσσ κκκκ

● Letting : =A

∫ ∫−T

t

v

tC

svC sdzedveVar )(*κκσ

∫ ∫ ∫∫

××= +−

T

t

T

t

v

tC

sv

tC

svvC sdzesdzeedvdvA

2

2

1

121 )()( 2*

1*)(

21κκκσ

⇒ A dv dv e e dsCv v s

t

v v

t

T

t

T

= ×

− + ∫∫∫ 1 221 2

1 2

σ κ κ( )min( , )

∫∫∫ =×),min(

2211

2121

)²()() )() :becausevv

t

v

t

v

t

dssfsdzf(ssdzf(s

Cerna 36


⇒ [ ] [ ]∫ ∫∫

−××

+−××

= +−+−

T

t

tvT

v

vvC

tvv

t

vvC eedveeedvedvA κκκκκκ

κσ

κσ 22

2)(22

2)(

11

1

212

1

21

21

21

⇒ ( ) ( )∫ ∫∫

−+

−= −−−−−−

T

t

T

v

vvtvvv

t

vvtvvC dveedveedvA1

2121

1

21122

)2()(2

)2()(1

2

2κκκκ

κσ

⇒ [ ]∫ −−−−−−− −++−−+−=T

t

vtvTtTvvtvtvtc eeeeeedvA )(2)2()()()(2)(12

2111111 11

2κκκκκκ

κσ

⇒ ( ) ( ) ( )

−+−−−−×= −−−− κτκτκτκτ

κκκκτσ eeeeA C

23332

2

211

2111

⇒ ( )κτκτκτ

κσ

κσ

κτσ −−− −+−−−= eeeHA CCC 2

3

2

2

2

2

2

12

⇒ ( )A HHC C= − − ×

−

τ

σκ

σκ

2

2

2

2²

with: κ

κτ−−=

eH 1

● Letting :

= ∫ ∫∫−

T

t

T

tSS

v

tC

svC ydzsdzedveCovB )(,)( ** σσ κκ

∫ ∫∫ ×

= −

T

t

T

tS

v

tC

svSC ydzsdzeedvB )()( **κκσσ

⇒ ∫ ∫ ∫

×= −

T

t

v

t

v

tSC

svSC sdzsdzeedvB )()( **κκσσ

with : [ ] dsdzdzE CS ρ=× **

⇒ B dv eS Cv s

t

v

t

T

= ×

− ∫∫( )σ σ ρ κ κe ds

Cerna 37


⇒ [ ]B eS C t v

t

T

= × −

−∫

σ σ ρκ

κ1 ( ) dv

⇒ [B S C= × − ]Hσ σ ρ

κτ

Thus Var[G(T)] can be written:

( )

−×−+

−×−−= )(2

2)]([ 2

22

2

2

HHHTGVar CSS

CC τκ

σρστσκ

σκστ

The futures price F(S,C,t,T) is the expectation, under the risk neutral probability, of

S(T). Therefore, we can write:

[ ])()]([),,,( TGQtt eETSETtCSF ==

⇒ )](var[

21)]([

)]([),,,(TGTGEQ

t

QteTSETtCSF

+==

( )

)(21

41)(

21)(

21)(ln)](var[

21)]([

2

22

2

22

H

HHtCHrtSTGTGE

CSS

CCS

Qt

−−+

−

−−−−−−+=+

τκ

σρστσ

κσ

κστατστατ

⇒ τκ

σκ

σρσκ

σατ rHtHCHtSTGTGE CCSCQt +−−

+−×−+=+

4)(

2)()(ln)](var[

21)]([

22

2

2

⇒ )exp())(exp(4

5.0²)(exp)(22

2

2](var[21)]([

τκ

σκ

σκρσσακτ rtCHHHtSe CCSCTGTGEQ

t

××−×

−

+−×−×=

+

⇒ ττ rtHC eeAtSTtCSF ×××= − )()()(),,,(

with : κ

κτ−−=

eH 1

−

+−×−=

κσ

κσκρσσακττ

4]5.0²[)(exp)(

22

2

2 HHA CCSC

Cerna 38


κλαα −=

This solution can also be written in the following way:

[ ])(exp)(),,,( τBHCtSTtCSF +−×=

with:

−×

−++

−×+

×

−×+−=

−−

2

2

3

22

2

2 1141

21][

κκσρσσκα

κστ

κρσσ

κσατ

κτκτ eerB CCSC

CSC

Volatility of futures returns

dFF F

F dt F dS F dC F dS F dC F dCdSt S C SS CC SC= + + + + +

1 12

12

2 2( ) ( ) ( ) ( ) ( )

Partial derivatives:

( )F S Ce B bt = × − + ×−κττ exp( )

with : κκ

σρσσκα

κσ

κρσσ

κσ

ακτκτ

τ

−−

×

−+++

−+−=

eerB CCS

CCSC2

2

22

2

2

22

F Ce

B bS = −−

+

=

−

exp ( ) exp( )1 κτ

κτ

FSS = 0

)exp()(1exp1 bHSBeCeSFC ××−=

+

−−×

−×−=

−−

τκκ

κτκτ

+

−−=

−=

−−

)(1 and 1with τκκ

κτκτ

BeCbeH

)exp()(1exp1

)exp(²)(1exp12

bHBeCeF

bHSBeCeSF

SC

CC

×−=

+

−−×

−−=

××=

+

−−×

−×=

−−

−−

τκκ

τκκ

κτκτ

κτκτ

Cerna 39


Noting that the futures price can be written: F S C t T S b( , , , ) exp( )= × , we obtain:

( )( ) dtHdtHdzHdtCHdzrdtBCeF

dFCSCCCSS σρσσσακστ

κτ −++−−+++−= − 22

21

Thus, we can write :

( )CCSS dzHdzVarF

dFVar σσ −=

Consequently, the term structure of volatilities is the following:

σ τ σ σκ κ

ρσ σκτ κτ

F S C Se e2 2 2

21

21

( ) = +−

− ×

−×

− −

C

When maturity tends towards infinity, the volatility of futures returns is:

limτ

σ σσκ

ρσ σκ→∞

= + −F SC S2 22

22 C

Cerna 40


Appendix 2.3. Analytical solution of the three-factor model

The first part of this appendix is devoted to the analytical solution of the three-factor




( ) ( )( )[ ] 0

21

21

21

3

2123321331122123

22

221

=−−−+

−−+−−++++++

τλ

λκλρσσρσσρσσσσσ

FFvva

FySFyvFFSFFFFS

v

ySyvSvSyvvyySS

with F(S, y, v, T, T )= S(T).

In order to solve this equation, we use a Feynman-Kac solution. In a risk-neutral world,

it is possible to consider the future prices F(S, y,v, t, T) as the expectation, at t and under the

probability Q, of the spot price at T:

)]([),,,,( TSETtvySF Qt=

Originally, the dynamics of the state variables is the following:

( )

( )

+−=+−=

+−=

33

22

11

dzdtvvadvdzydtdy

SdzSdtyvdS

σσκ

σ

with:

dtdzdz 1221 ρ=

dtdzdz 1331 ρ=

dtdzdz 2332 ρ=

Cerna 41


where:

- S is the spot price - y is the demeaned convenience yield, with y = C - α, where α is the long run mean

of the convenience yield C - v is the expected long-term spot price return; with v = µ - α, where µ is the drift of

S - κ is the speed of adjustment of the demeaned convenience yield - a is the speed of adjustment of v - v is the long run mean of the expected long-term spot price return - σi is the volatility of the variable i - ρij is the correlation between the variable i and j - dzi is the increment of a standard Brownian motion associated with the variable i

In the absence of arbitrage opportunity, this dynamic (changing λ2 in -λ2/κ) , under the risk neutral probability Q, is the following :

( )( )

( )

*1 1

*2 2

*3 3

dS v y Sdt Sdz

dy y dt dz

dv a v v dt dz

= − − λ + σ = κ λ − + σ

= − − λ + σ

1

2

3

with: dtdzdz 12

**21

ρ=

dtdzdz 13*3

*1 ρ=

dtdzdz 23*3

*2 ρ=

where λi is the risk premium for each of the three risk factors.

The value of the futures price can be determined from the well known result that under this risk neutral probability it is given by:

( , , , , ) ( ( ) | )tF t T S y E S Tν = F

Under this probability, we have also:

( ) ( )( ) ( )

σν − −λ − − +σ∫ ∫

=

2T T1u u 1 1 1t ty du T t dz

2S T S t e( )u

. (F)

νt and yt , are gaussian, as well as their integral. Now looking at the previous formula we see

that the drifts of yt , νt and λ1 add up. Thus, if we note

Cerna 42


2 1 3( )µ = λ + λ − ν − λ

we can re write the dynamics using new processes y and ν without drift:

( ) *1 1

*2 2

*3 3

dS v y Sdt Sdz

dy ydt dz

dv a dt dz

= − − µ + σ = −κ + σ

= − ν + σ

and again arrive to formula (F) ( with λ1 replaced by µ and y,ν given above). So that we have

to evaluate

( )( , , , , ) ( | )G Tt tF t T S y S E eν = F

where G(T) is gaussian.

A simple computation shows that G(T) (given Gt ) can be split into four terms:

( ) ( ) - ( , ) ( , ) ( )t tG T m y H k H a G T= τ τ + ν τ + 0

with :

( , )uv1 eH u v

u

−−=

21m( ) [ ]2

στ = − µ + τ

0 1 1 2 2 3 3( ) ( ) ( ) ( )T T T T T

ku kv au av

t t t t t

G T dz u e e dz v e e dz v− − − −= σ −σ +σ∫ ∫ ∫ ∫ ∫

so that we can write :

0( ) ( , ) ( , ) ( )( , , , , ) ( | )τ − τ +ν τν = t tm y H k H a G Tt tF t T S y S e E e F

where G0(T) is a zero mean gaussian. In order to compute it's variance it is convenient to write:

2 30 1 1 0 0G (T) z ( ) G (k) G (a)= σ τ − +

Cerna 43


with :

0( ) ( , ) ( )−= = σ∫ ∫ ∫T T v

i bvi i

t t t

G b Q b v dv e dv e dz sbsi

2 3 20 t 1 0 0

3 21 1 0 1 1 0

3 20 0

var(G (T) | ) Var(G (a)) Var(G (k))

+2 Cov(z ( ),G (a)) 2 Cov(z ( ),G (k))

-2 Cov(G (a),G (k))

= σ τ + +

σ τ − σ τ

F

from the computation of the two-factor model we get:

( ) ²( , )( ( )) ( ) ( ,− σ σ τ = σ = τ − τ × − ∫ ∫T v 2 2

i bv bs i i0 i i 2

t t

H bVar G b Var e dv e dz s H b2bb

10 1 1 1 1( ( ), ( )) ( ( ) , ( )) ( ( , )

T v Ti iji bv bs

i it t t

Cov G b z Cov e e dz s dv dz s H bb

− σ σ ρσ τ = σ σ = τ − τ∫ ∫ ∫ )

Thus, we just need to compute:

3 20 0 3 2 3 2( ( ), ( )) ( , ) , ( , ) ( ( , ), ( , ))

= = = ∫ ∫ ∫ ∫T T T T

t t t t

R Cov G a G k Cov Q a u du Q k v dv Cov Q a u Q k v dudv

We first find :

( ) ( )3 2 13

( )( ) ( )

3 2 13 [ ]

∧+ − +

− ++ ∧ +

= σ σ ρ

= σ σ ρ −+

∫ ∫ ∫

∫ ∫

T T u va k w av ku

t t tT T av ku

a k u v a k t

t t

R e dw e dudv

ee e dudva k

and finally :

3 2 23 1 1 1 1[ ( ) ( , ) ( , ) ( , ) ( , )]σ σ ρ= τ + − τ − τ − τ τ

+R H k H a H

a k a k k aa H k

Cerna 44


So we can write the future in the following condensed form:

t ty H(k, ) H(a, ) ( )tF(t,T,S, , y) S e− τ +ν τ +ϕν = τ

with :

( , ) ( , ) ( , ) ( , )( )

( ( , )) ( ( , ))

[ ( ) ( , ) ( , ) ( , ) ( , )]

2 22 23 22 2

1 3 12 1 2 12

3 2 23

H a H a H k H k2a 2ka k

22 H a H ka k

1 1 1 12 H k H a H a H ka k a k k a

τ − τ τ τ − τ τϕ τ = −µτ + σ − + σ −

σ σ ρ σ σ ρ

+ τ − τ − τ − τ

σ σ ρ+ τ + − τ − τ − τ τ

+

So that except for the value of µ, under the historic or risk neutral probability, there is

no difference between the dynamics and the future's value.

2. Volatility of futures returns

The volatility of futures returns is the following:

( ) ( ) ( ) ( ) ( ) ( )( )κ

ρσσρσσκ

ρσσσκ

σστστκττκττκτ

aee

aee

aee aaa

F

−−−−−− −−−

−+

−−

−+

−+=

1121212112332133112212

2232

222

21

2

which converges, as τ goes to infinity, to:

( )κ

ρσσρσσκ

ρσσσκσστσ

aaaF233213311221

2

23

2

222

12 222

−+−++=∞→

This volatility term structure decreases with maturity and converges to a positive constant. This is consistent with a mean reverting non stationary process.

Cerna 45


Part 3. Dynamic hedging strategies

Two important applications have been considered for term structure model of commodity prices: dynamic hedging strategies and investment decisions. The study presented in this report is centred on the first of these two applications.

The first application of term structure models is the hedge of long-term commitments on commodity markets, when “naïve” or traditional hedging strategies are neglected. Naïve strategies suggest taking a short (long) position in the futures market in order to hedge a long (short) position in the physical market, the two positions having the same size and expiration date. These strategies cannot be initiated when the position to hedge has a horizon superior to the exchange-traded contracts. Whereas this kind of problem has been tackled by Ederington [1979], it acquires a new dimension with term structure models. Indeed, these models rely on arbitrage reasoning and on the construction of a hedge portfolio. Therefore, their elaboration leads naturally to the study of hedging strategies.

Reflection on the use of term structure models for hedging purposes was motivated by Metallgesellchaft’s experiment6. At the beginning of the 1990s, the firm tried to cover long-term forward commitments on the physical market with short-term futures contracts. This attempt ended in a resounding failure: USD 2.4 billion were lost. However, it initiated research, the goal of which was knowing whether such an operation could be safely undertaken.

The interest of the Metallgesellschaft case is twofold. Firstly, being able to have long-term positions on the physical market while hedging the product of such sales constitutes an important stake. The success encountered by Metallgesellschaft when it proposed forward sales for a horizon of 5 or 10 years is indicative of that point. Secondly, Metallgesellschaft tried to hedge its long-term commitments in the physical market with short-term positions in the futures market. This kind of strategy is particularly interesting when the maturity of actively traded futures contracts is limited to a few months.

In this part of the report, we will show what conditions must be fulfilled in order to safely hedge a long term commitment on the physical market with short term positions in the futures market. The hedging problem will be separated into two steps. First of all, we will consider that the delivery date of the commitment on the physical market and the expiry date of the futures

6 For more information on the Metallgesellschaft case, see for example Culp and Miller ([1994] and [1995]), Edwards and Canter [1995].

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contract are the same. This will enable us to present the basic principles of traditional hedging (3.1). Then, we will introduce a gap between the maturity of the positions on the physical and the futures market. This will enable us to show how the hedging positions are modified when dynamic hedging is undertaken. In order to simplify the presentation, margin calls and deposits are ignored (3.2.). Lastly, we will present our analysis framework (3.3.).

3.1. Basic principles of traditional hedging

In order to present the basic principles of traditional hedging, let us take an example.

Suppose that, at a date t, an operator sells forward a barrel of crude oil for delivery in T. This operator does not have this barrel now. He decides that he will wait until a few days before the delivery, in T, and then buy the barrel on the spot market. This strategy avoids the operator to sustain the storage costs associated with the barrel of crude oil between t and T, but it also creates the risk of a rise in spot prices. In order to protect his profit on the physical market, the operator must take a position in the futures market. In the case of a traditional hedge, the values of the positions on the physical and the paper markets must be identical. Moreover, the position initiated in the futures market must be symmetrically opposed the position hold in the physical market. This will ensure that the losses recorded in one market will be compensated by a gain in the other market.

At t, in order to determine the price of his forward sale, the operator will use the futures price for delivery at T. Let us suppose, in order to make it simple, that the forward price equals F(t,T). In other words, we suppose that interest rates are constant7 and that there is no commercial profit associated with the forward sale. This profit could be easily introduced in the analysis.

Being exposed to a rise in the spot price, the operator buys a futures contract. The latter must be for delivery at T and it must represent a volume of one barrel of crude oil. The futures price of this contract is F(t,T). The financial result of this operation is the following:

At t: Buy one futures contract of maturity T, whose price is F(t,T) Sell forward a barrel deliverable at T, and fix the price at F(t,T) At T: Sell the futures contract of maturity T, whose price is F(T,T) = S(T)8 Buy on barrel of crude oil on the spot market at S(T)

7 Cox, Ingersoll, Ross (1981). 8 We suppose, in this example, that there is a perfect convergence of the basis at expiration: at maturity, the futures price of the nearby contract and the spot price are equal, otherwise there would be arbitrage operations. However, the convergence is never perfect in the real world, because arbitrage operations are costly. Thus, it is possible to sometimes observe a small difference between the spot and the futures price.

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The global result of the operation can be separated into two parts: Result on the physical market : F(t,T) – S(T) Result on the futures market : -F(t,T) + F(T,T) Global result : F(t,T) – S(T) - F(t,T) + F(T,T) = 0

Whenever the operator chooses the futures prices F(t,T) in order to value his forward sale, and provided that there is a perfect convergence of the basis at maturity, it suffices to buy a futures contract of maturity T in order to guarantee a perfect hedge. Thanks to this operation, even if he does not know what will be the spot price in T, the operator can fix his profit as early as at t. However, this result is guaranteed only if the operator and his counterpart on the physical market keep their respective commitment. There is indeed a risk for the operator that, between t and T, his counterpart on the physical market goes bankrupt. More likely, the spot price could evolve, between t and T, in such a way that the counterpart on the physical market requires a second negotiation of the sell price. The more the date T is far-off, the more these risks rise.

3.2. Dynamic hedging: presentation

Let us now consider the same hedging problem, and let us introduce a temporal gap between the positions on the physical and the paper market. Such a strategy can be motivated by the lack of liquidity on the longer maturities, which appears in most commodity markets9, and/or by the fact that the available futures contracts do not have a sufficient maturity to cover the position hold on the physical market (as was the case for Metallgesellschaft).

3.2.1. The basic principles of dynamic hedging

At a date t, an operator sells forward a barrel of crude oil for delivery in T. This operator does not have this barrel now. He decides that he will wait until a few days before the delivery, in T, and then buy the barrel on the spot market. To determine the price of his forward sale, if there is no contract for delivery at T, the operator will have no information provided by the futures market. In that case, he must use a term structure model, which gives the possibility to compute a futures price for any delivery date.

In order to protect his profit on the physical market, the operator must also take a position in the futures market. In the case of a dynamic hedge, the maturity of the position on the futures market will be different of that of the physical commitment. Thus, there is a need to determine the proper hedging ratio. The hedging period, between t and T, must indeed be decomposed into several sub-periods. At each sub-period, a new position on the futures market must be determined. This position must have the same value as the position on the physical market and must be symmetrically opposed to it. 9 In the American crude oil market, in 2003, more than 81% of the trading volume was concentrated on the three first months! (Lautier and Riva, 2004).

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Let us suppose that, for liquidity concerns, the futures position is the shorter one. The use of shorter maturities supposes that the hedge portfolio is rebalanced regularly as the future’s expiration date approaches, in order to maintain constantly the position on the paper market. Thus, this strategy exposes to a rollover basis risk that must be taken into account.

In the case of a dynamic strategy, the contribution of a term structure model is twofold. First, it authorizes the valuation of the commitment on the physical market for any delivery date. Second, it enables the determination of the hedge ratios. The more the model will be able to replicate the term structure, the more efficient the hedging strategy will be, and the closer to a traditional strategy it will be.

3.2.2. Dynamic hedging strategies and term structure models

Term structure models are naturally well suited for hedging strategy, because their construction relies on arbitrage reasoning and the elaboration of a hedge portfolio. Moreover, as was presented in the first part of this report, the term structure models rely on the identification of the underlying factors that explain the evolution of the futures price. These factors represent the uncertainty sources affecting the futures price. Thus, in this theoretical framework, the hedge strategy will be constructed in order to protect the operators against each of these uncertainty sources. This implies that the number of futures contracts required for the hedge is equal to the number of factors in the model used. Moreover, to properly hedge a forward commitment, the sensitivity of the present value of the commitment with respect to each one of the underlying factors must equal the sensitivity of the portfolio of futures contracts used to hedge the commitment, with respect to the same factors.

The literature on term structure models and dynamic hedging shows that the various hedging strategies based on term structure models differ mainly from each other in the assumptions concerning the behaviour of the futures prices.

In 1997, Brennan and Crew compared the hedging strategy initiated by Metallgesellschaft on the crude oil market with other strategies relying on the models of Brennan and Schwartz [1985] and of Gibson and Schwartz [1990]. The authors showed that the strategies relying on the term structure models outperform by far that of the German firm all the more, as the term structure model is able to correctly replicate the prices curve empirically observed.

The same year, Schwartz also calculated the hedge ratios associated with each of the three models he studied. However, he did not test the efficiency of the related hedging strategies.

In 1999, Neuberger compared the performances of hedging strategies relying on the two-factor model developed by Schwartz in 1997 and on a new model. In this new model, no assumption is made on the number of variables, about the process they follow, or about the way risk is priced. The key assumption is that the expected price at which the long-dated contract

Cerna 49


starts trading is a linear function of the price of existing contracts (whereas in Schwartz’ model, the futures price is a non-linear function of the state variables). While theoretically inconsistent with some models of the term structure of commodity prices, in practice this new model still gives good results. With this model, the author constructed hedge portfolios based on an arbitrary number of futures contracts with different maturities. His strategy was quite successful in eliminating most of the risk exposure (approximately 85% as measured by the standard deviation of the hedge error) in the crude oil market. The author believes that the robustness of his model derives from the paucity of assumptions. The drawback of hedging strategies based on term structure models is that the latter are efficient only if futures prices are fairly priced relative to each other. However, Neuberger underlines that his model requires balancing the portfolio a bit more frequently than Schwartz’.

In 2000, Routledge, Seppi and Spatt showed that hedge ratios based on their term structure model are not constant, but rather conditional on the current demand shock and the endogenous inventory level. These ratios take into account the fact that short-term and long-term forward prices differ, in that long-term prices do not include the option to consume the good between the two delivery dates.

Lastly, in 2003, Veld-Merkoula and de Roon used a one-factor term structure model based on convenience yield to construct hedge strategies that minimize both spot price risk and rollover risk by using futures of two different maturities. They take into account the transaction costs associated with their strategy, which outperforms the naïve hedging strategy. However, the authors do not compare their results with previous works.

This brief review of the literature shows that little work has been done on dynamic hedging. Indeed, empirical work is scarce. The authors studied the hedge ratios but, except Brennan and Crew, Neuberger, and Veld-Merkoulava and de Roon, they did not pay attention to the hedge strategies themselves. In this report, we choose to apply the framework of Brennan and Crew, because it is the more complete and interesting. Indeed, Neuberger is more interest in empirical than in theoretical issues, and the framework of Veld-Merkoulova and de Roon is too specific. We will first of all apply the framework of Brennan and Crew to the term structure models selected in the second part of the report. These models are indeed more interesting than those chosen by the two authors, simply because researches have progressed since 1997. Then, because until now, little work has been done on transaction costs and on the date of the rebalancing of the portfolio, we will study the impact of these variables on the efficiency of the hedging strategies. Lastly, we will compare the results obtained with the term structure models with those obtained with a purely empirical model.

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3.3. Dynamic hedging: the analysis framework

We first of all recall of the commitment of the operator on the physical market. Then, we present the dynamic hedging strategies and the hedge ratios associated with the three term structure models that were retained. Lastly, we discuss the way we will study the efficiency of these dynamic hedging strategies, and we expose the concept of hedging error.

3.3.1. The commitment

At a date t, an operator sells forward a barrel of crude oil for delivery in T. This operator does not have this barrel now. He decides that he will wait until a few days before the delivery, in T, and then buy the barrel on the spot market. This strategy avoids the operator to sustain the storage costs associated with the barrel of crude oil between t and T, but it also creates the risk of a rise in spot prices.

3.3.2. The hedging strategies associated with the three term structure models

In order to protect his commitment against fluctuations in the spot price, the operator initiates a hedging strategy on the futures market. The maturity of the position hold on the paper market is shorter than the one of the commitment, leading the operator to undertake a stack and roll strategy. Indeed, the use of shorter maturities supposes that the hedge portfolio is rebalanced regularly as the future’s expiration date approaches, in order to maintain constantly the position on the paper market. The value of the hedge position must be identical to the present value of the commitment. This will ensure that the losses recorded in one market will be compensated by a gain in the other market.

The hedging strategy is approached using the term structure models of commodity prices selected in the second part of the report. Thus, it relies on the relationship between the futures prices of different maturities and it uses combinations of futures contracts having different maturities. The number of futures positions is equal to the number of underlying factors (i.e. state variables) included in the term structure model. Thus, each term structure model has specific implications for hedging strategies, because each model contains a different number of underlying factors, which will lead to different hedge ratios. However, since the three models assume constant interest rates, and therefore that futures prices are equal to forward prices, the present value of the forward commitment per unit of the commodity can simply be obtained by discounting the future (forward) price10.

To properly hedge the forward commitment, the sensitivity of the present value of the commitment with respect to each one of the underlying factors must equal the sensitivity of the 10 Actually, the three-factor term structure model allows for non constant interest rates. However, in order to simplify our study and to make the models comparable, we choose to work with constant interest rates.

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portfolio of futures contracts used to hedge the commitment, with respect to the same factors. Thus, the number of positions wi in futures contract with maturity τi required to hedge the forward commitment to deliver one unit of a commodity at time T is obtained, in each one of the models, by solving a system of equations.

The first model is a one-factor model. Consequently, it requires only one position in the futures market:

( ) ( )TSFetSFw SrT

S ,, 11−=

The position in the futures market represents here the uncertainty associated with the spot price. Thus, it seems natural to choose a short term contract.

The second model contains two factors, leading to the determination of two positions for delivery in t1 and in t2:

( ) ( ) ( )( ) ( ) (

=+

=+−

−

TCSFetCSFwtCSFwTCSFetCSFwtCSFw

CrT

CC

SrT

SS

,,,,,,,,,,,,

2211

2211

)

The positions in the futures market represent the uncertainty associated with the spot price and with the convenience yield. There is no traded asset representing the convenience yield. This is the reason why a second position in the futures market is taken as a hedge against this uncertainty source.

Lastly, the three-factor model requires three positions in the futures market, respectively representing the spot price, the convenience yield, and the long term price:

( ) ( ) ( ) ( )( ) ( ) ( ) (( ) ( ) ( ) (

=++

=++

=++

−

−

−

TvySFetvySFwtvySFwtvySFw

TvySFetvySFwtvySFwtvySFwTvySFetvySFwtvySFwtvySFw

vrT

vvv

yrT

yyy

SrT

SSS

,,,,,,,,,,,,

,,,,,,,,,,,,,,,,,,,,,,,,

332211

332211

332211

))

The examination of these systems shows that all these hedge ratios are state dependent and they decline with the maturity of the forward position. Moreover, in order to determine the weight wi, it is necessary to choose arbitrarily the maturity τi.

The solutions of these linear systems can be found in Appendix 3.

3.3.3. The efficiency of the hedging strategies

The efficiency of the hedging strategies is appreciated on the basis of a comparison between the result of the hedging strategy and the return obtained on a risk-less loan. The hedging strategy is all the more efficient that it is close to a risk-less operation.

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3.3.3.1. The hedging error

In the case of dynamic strategies, the hedging period is decomposed into several sub-periods, and new hedge ratios must be computed each time the position is renewed on the futures market. Thus, an intermediate hedging error corresponding to each sub-period can be computed, and the total hedging error corresponds to the sum of these intermediate errors.

In order to define such an intermediate error, suppose, for example, that the hedging positions are rolled each month. This means that, each month, an intermediate hedging error can be calculated, on the basis of a comparison between the intermediary result of the hedge position and the result of a risk-less loan of the present value of the forward commitment, during one month.

Let us first of all concentrate on the intermediary result of the hedge position at a date (t+1), between t and (t+1).

In order to value, at a date (t+1) – one month after the beginning of the forward commitment – the intermediary hedging error corresponding to a strategy S and a delivery date T, the present value of the commitment L is first of all calculated:

)1,1(1,1

1,1 −+= −+−−+ TtFeL tr

tτ

τ

where:

- L is the present value of the commitment, - r is the interest rate, - F is the futures price, - τ is the maturity of the commitment,

In order to make a distinction between the prediction errors of the term structure models and the errors associated with the hedge ratios – which are the only interesting errors in our framework – F(t+1,T-1) is the futures price observed on the market at the date (t+1) for delivery at (T-1).

The variation of the present value of the commitment between t and (t+1) represents the intermediary result that the operator obtains on the physical market:

1,1,, −+−=∆ τττ ttt LLL

Let us now consider the intermediate result on the futures market. Suppose that the hedge position is made of two different positions w1 and w2, whose maturities are respectively τ1 and τ2. In that case, the intermediate result recorded during one sub-period on the futures market can be written:

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2211 FwFw ∆+∆

where ∆Fi represents the variation of the futures price of maturity τi between t and (t+1):

),()1,1( iii tFtFF ττ −−+=∆

Naturally, the pairs (wi,τi) change with the model and with the maturity sets chosen for the hedge ratios.

The variation in the wealth of the operation between t and (t+1) can be now computed. This variation depends on the variation of the present value of the forward commitment and on the losses or profits generated on the futures market. If we call Wt,τ the wealth of the operator at t, the variation of this wealth between t and (t+1) can be written:

ττ ,2211, tt LFwFwW ∆−∆+∆=∆

Lastly, it is possible to determine the intermediate hedging error at a date (t+1). This error Et,τ relies on the comparison of the intermediate result on the hedged operation, ∆Wt,τ , and the loan, during one sub-period, of a value corresponding to the forward commitment on the physical market:

t, t, t, t,1E W L rτ τ τ= ∆ −

For each strategy, the total hedging error corresponds to the sum of the intermediate errors.

3.3.3.2. Hedging error with transaction costs

In this framework, it is easy to introduce transaction costs. We suppose that there are transaction costs in the crude oil futures market as well as in the interest rate market. In the latter case, however, the transactions costs considered is the credit spread which is imposed to the lender or the borrower, as a function of his rating.

In a first approach, we make the assumption that transaction costs are directly proportional to the amount invested in a market. The intermediate hedging error can then be written in the following way:

t, t, H t, t,1 LE W L (rτ τ τ = ∆ − ϖ − − ω )

where Hϖ is the transaction cost on the hedged position, and Lϖ is the so called transaction cost on the interest rate market.

This framework gives the possibility to undertake simulations with different relative levels of transactions costs. It gives also a mean to study the following alternative:

- diminution of the length of the sub-period and of the hedging error and rise in the transaction costs

- rise of the length of the sub-period and of the hedging error and diminution of the transaction costs.

The empirical tests on these dynamic hedging strategies will be undertaken in the next part of this report.

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Appendix 3. Hedge ratios associated with the three term structure models

The number of positions wi in futures contract with maturity τi required to hedge the forward commitment to deliver one unit of a commodity at time T is obtained, for each model, by solving a system of equations.

1. The one-factor model

The first model is a one-factor model. Consequently, it requires only one position in the futures market:

( ) ( )TSFetSFw SrT

S ,, 11−=

Writing:

( )( , )SYF S

Sτ

τ =

we find:

)()(

1τττ

YYew r−=

where:

( ) ( )2

2ˆ( ) exp ln 1 14

Y e e S e e−κτ −κτ −κτ − κτ στ = + α − + − κ

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2. The two-factor model

The second model contains two factors, leading to the determination of two positions for delivery in t1 and in t2. The linear system to solve is the following:

( ) ( ) ( )( ) ( ) (

=+

=+−

−

TCSFetCSFwtCSFwTCSFetCSFwtCSFw

CrT

CC

SrT

SS

,,,,,,,,,,,,

2211

2211

)

Remember that the solution of this model is :

( , )( , , ) ( ) H k C rF S C S A e e− τ ττ = × τ × ×

writing : ( , , ) ( )F S C SYτ = τ

we find :

( , , ) ( )( , , ) ( , ) ( )

S

C

F S C YF S C H k SY

τ = τ

τ = − τ τ

And the linear system can be written in the following way:

H w bΛ =

with:

1 1 ( )

( , ) ( , ) ( )( )

r 1

1 2

Y 0eHH k H k 0 YY

τ τ = Λ = − τ − τ ττ 2

and:

( , )1

bH k

= − τ

Letting W , we have to solve w= Λ HW b= and we find:

( , ) ( , )( , ) ( , )( , ) ( , )

( , ) ( , )

22 1

12 1

H k H kH k H k

WH k H k

H k H k

τ − τ τ − τ =

τ − τ τ − τ

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then, multiplying by the inverse of Λ we obtain:

( , ) ( , )( ( , ) ( , )) ( )

( )( , ) ( , )

( ( , ) ( , )) ( )

22 1r

12 1

H k H kH k H k Y

w e YH k H k

H k H k Y

− τ

τ − τ

1

2

τ − τ τ = τ

τ − τ τ − τ τ

Note that in the preceding formula we can replace Y ( )τ by and Y by , showing that each weight depends upon a ratio in the futures values and a function

depending only of the maturities and the factor controlling the reversion of the convenience yield.

( , , )F S C τ ( )iτ( , , )iF S C τ

3. The three-factor model

In the case of the three factor model, the system to solve is the following:

( ) ( ) ( ) ( )( ) ( ) ( ) (( ) ( ) ( ) (

=++

=++

=++

−

−

−

TvySFetvySFwtvySFwtvySFw

TvySFetvySFwtvySFwtvySFwTvySFetvySFwtvySFwtvySFw

vrT

vvv

yrT

yyy

SrT

SSS

,,,,,,,,,,,,

,,,,,,,,,,,,,,,,,,,,,,,,

332211

332211

332211

))

If we write ( , , , ) ( )F S y SYν τ = τ

we have:

( , , , ) ( )( , , , ) ( , ) ( )

( , , , ) ( , ) ( )

S

y

F S y YF S y H k SY

F S y H a SYν

ν τ = τ

ν τ = − τ τ

ν τ = τ τ

And the linear system can be writen:

H w bΛ =

with:

( ) ( , ) ( , ) ( , ) ( )

( )( , ) ( , ) ( , ) ( )

1r1 2 3 2

1 2 3

1 1 1 Y 0eH H k H k H k 0 Y 0

YH a H a H a 0 0 Y

τ τ = − τ − τ − τ Λ = τ τ τ τ τ τ 3

0

and:

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( , )( , )

1b H k

H a

= − τ

τ

Letting W , we have to solve w= Λ HW b= :

After some tedious computation, we find:

( , ) ( , ) - ( , ) ( , ) ( , )( ( , ) - ( , )) ( , )( ( , ) - ( , ))

( , ) ( , ) - ( , ) ( , ) ( ( , )( ( , ) ( , )) ( , )( ( , ) ( , ))W=

H( , ) ( ,

2 3 3 2 2 3 3 2

1 3 3 1 1 3 3 1

2

H k H a H k H a H k H a H a H a H k H kD

H k H a H k H a H k H a H a H a H k H kD

k H a

τ τ τ τ + τ τ τ + τ τ τ

τ τ τ τ + τ τ − τ + τ τ − τ

τ τ ) - H( , ) H( , ) +H( , ) ( ( , ) - ( , )) H( , )( ( , ) - H( , ))1 1 2 2 1 1 2k a k H a H a a H k kD

τ τ τ τ τ + τ τ τ

with:

( ( , ) - H( , )) H( , ) (H( , ) - H( , )) ( , ) +(H( , ) - H( , )) H( , ) 3 2 1 1 3 2 2 1D H k k a k k H a k k a= τ τ τ + τ τ τ τ τ τ3

and Finally:

( )

( )( )

( )

11

r 22

33

WYWw e Y

YW

Y

− τ

τ

= τ τ

τ

We can make the same remark than for the two-factor model.

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Part 4. Empirical tests

This part of the report aims to compare the dynamic hedging strategies associated with the different term structure models. For each model presented in the second part of the report, it will first of all measure the efficiency of dynamic hedging strategies. Then, it will study the efficiency as a function of the choice of a specific maturity for the hedging position.

We present first of all the data, then the method used for the estimation of the models, and lastly the tests on the dynamic hedging strategies.

4.1. The data

The database is a powerful tool for the study because it is very complete. It is divided into two parts: crude oil futures prices, and US interest rates. They represent a large volume of information, especially because we have the curves of crude oil prices and interest rates: it represents around 50 000 futures prices, and 70 000 interest rates.

The crude oil are daily settlement prices for the light, sweet, crude oil contract of the New York Mercantile Exchange, from February 1997 to February 2002 (5 years). They have been arranged such as the first futures price’s maturity corresponds actually to the one month’s maturity, such as the second futures price’s corresponds to the two months maturity, and so forth... Indeed, we reconstructed the Nymex calendar, in order to determine when a specific contract has, for example, a one- or a two-month maturity and to determine when the contract falls from the two-month to the one-month maturity. We reconstituted thus prices curves with monthly maturities from the first to the 28th months, and with early maturities from the 3rd to the 5th years. The daily interest rates are issued from US government bonds, and the data are available for every month, from the first to the 72th, between February 1997 and February 2002. These data were extracted from Datastream. Our last work was to render homogeneous the databases of crude oil prices and interest rates.

Figure 4.1 gives an overview of the behavior of the futures prices for a few selected maturities, during the period of observation. This period is especially interesting for several reasons. First, prices strongly fluctuate, ranging from a lower level of 11$/b to the highest level

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of 37$/b. It is thus particularly interesting to study hedging strategies on this kind of period. Second, the period is characterized by an alternation of contango and backwardation, giving us the possibility to study to types of configurations. The backwardation is however the most frequent situation, as was expected.

Figure 4.1. Observed futures prices, 1997-2002

10,00

15,00

20,00

25,00

30,00

35,00

20/0

3/19

97

20/0

7/19

97

20/1

1/19

97

20/0

3/19

98

20/0

7/19

98

20/1

1/19

98

20/0

3/19

99

20/0

7/19

99

20/1

1/19

99

20/0

3/20

00

20/0

7/20

00

20/1

1/20

00

20/0

3/20

01

20/0

7/20

01

20/1

1/20

01

($/b)

1 month

6 months

12 months

18 months

36 months

4 years

5 years

Figure 4.1 gives also an illustration of one of the most important features of the commodity prices curve’s dynamic: the difference between the price behaviour of first nearby contracts and deferred contracts. The movements in the prices of the prompt contracts are large and erratic, while the prices of long-term contracts are relatively still. This results in a decreasing pattern of volatilities along the prices curve. Indeed, the variance of futures prices and the correlation between the nearest and subsequent futures prices decline with maturity. This phenomenon is usually called “the Samuelson effect”. Intuitively, it happens because a shock affecting the nearby contract price has an impact on succeeding prices that decreases as maturity increases (Samuelson [1965]). As futures contracts reach their expiration date, they react much stronger to information shocks, due to the ultimate convergence of futures prices to spot prices upon maturity. These price disturbances influencing mostly the short-term part of the curve are due to the physical market, and to demand and supply shocks.

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4.2. Estimation of the models

In order to test the dynamic hedging strategies associated with the term structure models, there is first of all a need for the values of the parameters of each model. In this paragraph, we present the method used for the estimation of the parameters, the optimal parameters that we found, the performances of the models and the performances criteria that we used in order to measure these performances.

4.2.1. The method of the Kalman filter

The parameters values are necessary to compute the estimated futures prices and to compare them with empirical data. The parameters estimation is not obvious though, because the three term structure models rely on non-observable state variables. As it was explained in the first part of the report, the spot price, the convenience yield and the long-term price are indeed regarded as non-observable because most of the time there are no reliable time series for the spot price and the convenience yield and the long-term price are not traded assets. In order to deal with this difficulty, the method of the Kalman filter is used11.

A Kalman filter can be used for the estimation of a model’s parameters, when the model

relies on non observable data. The basic principle of the filter is indeed the use of temporal

series of observable variables to reconstitute the value of the non observable variables. The

Kalman filter is also an interesting method when a large volume of information must be taken

into account, because it is very fast. Last but not least, when associated with an optimization

procedure, the filter provides a mean to obtain the model’s parameters.

The method requires first of all the model to be expressed on a state-space form. A

measurement equation and a transition equation characterize a state-space model12. Once this

has been made, a three step iteration process can begin.

The transition equation describes the dynamic of the variables α~ , for which there are no

empirical data. This equation gives the only available information for these variables. It allows

the calculation of α~ in t, conditionally to their value in (t-1). Once this calculus has been made,

it is possible to obtain, via the measurement equation, the measure y~ in t. This second equation

represents the relationship linking the observable variables y~ with the non observable α~ . The

difference, in t, between the measure y~ and the empirical data y represents the innovation v.

Finally, this innovation is used to obtain the value of α~ at the date t, conditionally to the

11 These methods were extensively studied in the previous report written for the IFE (Lautier and Galli, 2002). This is the reason why, in the present report, we are not giving details.

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information available in t.

Thus the Kalman filter allows for the calculation of α~ , and updates its value when

some new information arrives. There is one iteration for each observation date t, and one

iteration includes three steps.

During the first step, the prediction phase, the values of the non observable variables in

(t-1) are used to compute their expected value in t, conditionally to the information available in

(t-1). The predictions rely on the transition equation. The predicted values 1/~

−ttα are then

introduced in the measurement equation to determine the measure . In this equation, the

errors have zero mean and are not serially nor temporarily correlated. They represent every kind

of disturbances likely to lead to errors in the data. The second step or innovation phase allows

for the calculation of the innovation v

ty~

t. Lastly, the values of the non observable variables, which

where calculated in the prediction phase, are updated conditionally to the information given by

vt. Once this calculus has been made, tα~ is used to begin a new iteration.

When the Kalman filter is applied to term structure models of commodity prices, the

objective is the estimation of the measurement equation’s parameters, in order to obtain

estimated futures prices for different maturities ( )iF τ~ , and to compare them with empirical

futures prices ( )iF τ . The closest the firsts are with the seconds, the best is the model. The way

usually chosen to appreciate the model’s performances is to analyze the estimation’s error for

one specific maturity τi and for the whole estimation period.

The application of the Kalman filter to the three term structure models chosen for the

empirical tests and the method used for the parameters estimation are presented in appendix. All

the problems usually encountered in the applications of the Kalman filter, (initialization of the

iterative process, stabilization of the iterative process, performances measures with a simple

Kalman filter) were described in the previous report on term structure models written in 2002

for the IFE. In the present report, we used the same methods and we applied them to the three

term structure models, which were chosen in part two. For the three models, it was possible to

use the simple Kalman filter, because all these models can be linearized just by taking logs.

12 There is more than one state-space form for certain models. Then, because some of them are more stables compared with the others, the choice of one specific representation is important.

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4.2.2. The optimal parameters

In order to estimate the optimal parameters associated with each term structure model,

we needed weekly observations on crude oil futures prices. Thus, keeping the first observation

of each group of five, we transformed our daily data into weekly data. Moreover, for the

parameters estimations, and for the measure of the models performances, four series of futures

prices were retained, corresponding to the one, the three, the six and the nine months maturities.

Lastly, because interest rates are supposed to be constant in our study, we used the mean of all

the observations having a three months maturity between 1997 and 2002.

Table 4.1, 4.2 and 4.3. represent the optimal parameters obtained for each term structure

model.

Table 4.1. Optimal parameters, one-factor model, 1997-2002 Parameters Gradients

Pull back force : κ 0,314794 -0,00683 Trend : µ 0,321326 0,000798

Spot price’s volatility : σS 0,425617 -0,00021 Risk premium : λ 0,811274 0,00239

Table 4.2. Optimal parameters, two-factor model, 1997-2002 Parameters Gradients

Pull back force : κ 1,51 0,0001 Spot price’s volatility : σS 0,285 -0,0023

Convenience yield’s volatility : σC 0,278 0,0015 Correlation coefficient : ρ 0,95 -0,1100

Long run mean : ν 0,007 0,0080

Table 4.3. Optimal parameters, three-factor model, 1997-2002

Parameters Gradients Pull back force : κ 0,78 0,0020 Pull Back force: a 6,78 0,0002

Spot price’s volatility : σ1 0,1 0,0610 y volatility's : σ2 0,31 0,0520 ν’s volatility : σ3 0,1 0,0460

Correlation coefficient : ρ12 0,41 -0,0009 Correlation coefficient : ρ32 0,28 -0,0035 Correlation coefficient : ρ23 0,53 0,0021

Average of ν -0,02 -0,0013

We can observe that these parameters change significantly with the models, as was

previously observed by Schwartz, for example. Moreover, their values are in line with previous

studies on the crude oil market.

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4.2.3. The performance criteria

The model’s performances measure its ability to reproduce the term structure of commodity prices. To assess these performances, two criteria were retained: the mean pricing error and the root mean squared error.

The mean pricing error (MPE) is defined as follows:

( ) ( )( )∑=

−=N

n

nFnFN

MPE1

,,~1 ττ

where N is the number of observations, ( )τ,~ nF is the estimated futures price for a maturity τ at date n and ( )τ,nF is the observed futures price. The MPE is expressed in US dollar. It measures the estimation bias for one given maturity. When the estimation is good, the MPE is close to zero.

Retaining the same notations, the root mean squared error (RMSE) is also expressed in US dollar and is defined in the following way, for one given maturity τ :

( ) ( )( )∑=

−=N

n

nFnFN

RMSE1

2,,~1 ττ

When there is no bias, the RMSE can be considered as an empirical variance. It measures the estimations stability. This second criterion is considered as more representative because prices errors can offset themselves and the MPE can be low even if there are strong deviations.

4.2.4. The models performances

Table 4.4, 4.5, and 4.6. reproduce the models performances on the observation period.

Table 4.4. Performances, one-factor model Maturity MPE RMSE 1 month -1,2953 4,6743 3 months -1,2756 3,9634 6 months -1,1337 3,1507 9 months -1,0100 2,5825 Average -1,1787 3,5927

Unit : USD/b.

Table 4.5. Performances, two-factor model Maturity MPE RMSE 1 month 0,042 1.43 3 months 0,175 1,17 6 months 0,141 0.91 9 months 0.034 0.78 Average 0,098 1,072

Unit : USD/b.

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Table 4.6. Performances, three-factor model Table 4.6. Performances, three-factor model Maturity Maturity MPE MPE RMSE RMSE

5

1 month -0.012 1.395 3 months 0.145 1.148 6 months 0.102 0.894 9 months -0.017 0.792 Average 0.055 1.057

Unit : USD/b.

A first global comment on these results is that they are in line with previous studies, as:

- the performances of the three-factor model are better than those of the two-factor model, with in return performs better than the one-factor model.

- the differences between the two- and three-factor models are slight. This confirms the comments on the interest of such a model that were made in the first part of the report (see 1.4).

- the performances increase with maturity. This phenomenon can be explained by the fact that the more maturity increases, the more futures price’s volatility diminishes.

A more specific comment can be made if we look at Figure 4.2. This figure displays the

errors for the three-factor model. First, we see that the errors are larger on the one-month future,

as was expected. Second, it is clear that the global performances become poorer in 1999 (week

100), when the price regime strongly changes (see Figure 4.1). As the same comment can be

done for the three models, we can infer that the performances of the dynamic hedging strategies

will probably decline on the end of our study period. In order to improve the accuracy of our

results, we could fit the models on smaller periods.

Figure 4.2. Residuals for the three-factor model, 1997-2002

W e e k s

Blue One month maturity , Green 3months,

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4.3. Dynamic hedging strategies

In this paragraph, we first present the strategies which were tested on the observation period. Then we present our empirical results.

4.3.1. The strategies

We choose three different delivery dates for the forward sale on the physical market. The maximum maturity is of 5 years, because the length of our database is also of 5 years. We selected the following maturities: 5 years, 3 years, and 18 months.

When the hedge portfolio is renewed each month, these choices for the expiration dates of the forward commitment gave us to possibility to study 67 strategies per term structure model and per sets of maturities for the hedge ratios:

- 1 strategy for a delivery in 5 years, from the 03/15/1997 to the 02/15/2002 - 24 strategies for a delivery in 3 years, from the 03/15/1997 to the 02/15/2000, from

the 04/15/1997 to the 03/15/2000, …. - 42 strategies for a delivery in 18 months, from the 03/15/1997 to the 09/15/1998,

from the 04/15/1997 to the 10/15/1998, etc.

For each of these strategies, it is then possible to retain different sets of maturities for the hedge ratios. Considering the fact that we have a three factors model, we need three positions on the futures markets, and then we must choose three different maturities. The following sets were selected: 2, 3, and 4 months, 2, 6, and 9 months, 2, 6, and 12 months for the three-factor model; (2,4), (2,6) and (2, 12) for the two-factor model, and (2), (6), and (12) for the one-factor model. These choices lead to 67 × 3 = 201 strategies per term structure model, and thus to 603 strategies.

For the choice of the maturities of the positions composing the hedge portfolio, we respected two different constraints. First, there is a need to keep the nearby contracts in the hedge portfolio, in order to benefit from the higher liquidity on these contracts. Remember that in the American crude oil futures market, the first three contracts account for 81.6% of the cumulated trading volume in 2003 (Lautier and Riva, 2004). This is why we kept the set of maturities of 2, 3, and 4 months. This set of maturities constitutes an extreme case, which is in the favour of liquidity. However, we had also to take into account the fact that each contract must represent a different source of uncertainty. If the positions in the hedge portfolio are too close from each other, the futures price will evolve in the same way and there is a risk that the hedge is not efficient. This is why we choose the maturities of 2, 6, and 12 months. We did not go further than 12 months, because almost 96% of the trading volume is concentrated on

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maturities below one year in the crude oil futures market. This set constitutes another extreme case, and the set of maturities of 2, 6, and 9 months is an intermediary case.

4.3.2. The empirical results

To better understand the results, we will first present the weights (hedge ratios) that we obtained for each hedging strategies. Then we will expose the performances of the strategies.

4.3.2.1. The weights of the hedging portfolios

In order to compute these weights, we used the optimal parameters that were previously determined for each model. We also needed the values of the unobservable state variables, namely St for the first model, Yt fo the second one and Yt,,Vt for the three-factor model. We obtained them by running the Kalman filter with the optimal parameters. We found thus the best estimate of each state variable, on a day by day basis.

The weights depend thus on the model and on the state variables. As the latter are defined by stochastic processes, they are varying according to their dynamic, and their trajectory is not regular although their estimate is smoother. This explains small irregularities on the weights.

The weights are presented for each maturity of the commitment on the physical market and for each model. Thus, Figures 4.3 to 4.5 correspond to the 5 years commitment, Figures 4.6 to 4.8 correspond to the 3 years commitment, and Figures 4.9 to 4.11 correspond to the 18 months commitment. In each case, the abscissa represents the time before maturity, which is expressed in months. The ordinate represents the value of the weights for a specific model. The blue line corresponds always to the shorter maturity retained for the portfolio (namely 2 months), the green is the second shorter maturity (namely 3, 6, or 9 months), and the red is the longer one. Moreover, Figures a (upper left) always correspond to the shorter maturity hedge, Figures b (upper right) correspond to the intermediate maturity hedge and Figures c (below) to the larger one.

For example, Figure 4.5.represents the weights of the three-factor model, which were determined in order to hedge a 5 years commitment on the physical market. The weights for a hedge with maturities (2,3,4) months are represented on Figure 4.5.a (upper left), the weights for a hedge with maturities (2,6,9) months are represented on Figure 4.5.b (upper right), and the weights for a hedge with maturities (2,6,12) are on figure 4.5.c (below).

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Figures 4.3.a, 4.3.b & 4.3.c. Weights for the one factor model when hedging a 5 years commitment

hedging portfolio: 2 months hedging portfolio: 6 months

hedging portfolio: 12 th

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Figures 4.4.a, 4.4.b & 4.4.c. Weights for the two factor model when hedging a 5 years commitment.

Hedging Portfolio: 2, 3 months



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Figures 4.5.a, 4.5.b & 4.5.c. Weights for the three factor model when hedging a 5 years commitment. Figures 4.5.a, 4.5.b & 4.5.c. Weights for the three factor model when hedging a 5 years commitment.

Hedging Portfolio: 2,6,12 months



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Figures 4.6.a, 4.6.b & 4.6.c. Weights for the one factor model when hedging a 3 years commitment Figures 4.6.a, 4.6.b & 4.6.c. Weights for the one factor model when hedging a 3 years commitment

Hedging portfolio: 12 months



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Figures 4.7.a, 4.7.b & 4.7.c. Weights for the two factor model when hedging a 3 years commitment Figures 4.7.a, 4.7.b & 4.7.c. Weights for the two factor model when hedging a 3 years commitment

Hedging portfolio: 2,12 months



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Figures 4.8.a, 4.8.b & 4.8.c. Weights for the three factor model when hedging a 3 years commitment Figures 4.8.a, 4.8.b & 4.8.c. Weights for the three factor model when hedging a 3 years commitment

Hedging portfolio: 2,6,12 months


Hedging portfolio:2,3,4 months

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Figures 4.9.a, 4.9.b & 4.9.c. Weights for the one factor model when hedging a 18 months commitment Figures 4.9.a, 4.9.b & 4.9.c. Weights for the one factor model when hedging a 18 months commitment




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Figures 4.10.a, 4.10.b & 4.10.c. Weights for the two factor model when hedging a 18 months commitment Figures 4.10.a, 4.10.b & 4.10.c. Weights for the two factor model when hedging a 18 months commitment

Hedging portfolio:2,12 months



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Figures 4.11a, 4.11.b & 4.11.c. Weights for the three factor model when hedging a 18 months commitment Figures 4.11a, 4.11.b & 4.11.c. Weights for the three factor model when hedging a 18 months commitment



Hedging portfolio:2,3,4 months

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The examination of the preceding Figures leads to the following comments.

- the one-factor model behaves quite well: the weights are positive and thus

correspond to a buy position on the future market as expected .They tend to zero for

large maturities. The higher value (2.4) is reached with a 12 month position on the

futures market. It corresponds to the 18 months commitment on the physical market,

one month before this position on the physical market expires.

- the absolute value of the weights are higher for the two and three-factor models. For

the two factor model one is positive and the other negative, except when the

residual maturity of the commitment to hedge is shorter than the highest hedging

maturity. For example, when hedging a 5 years commitment with the two factor

model (maturities 2 and 12 months), we see on figure 4.4.c that the weights

corresponding to residual maturities less than 12 and greater than 2 months are

positive. The weights pattern is even more complex in the three factor model: we

see many crossing for short residual maturities. This is clear on figure 4.5.c, when

hedging a 5 years commitment using futures with maturities 2,6,12 months. The

first one occurs at 12 months –counting back from 60 months-, this correspond to

the higher maturity of the hedging portfolio. Looking at the set of linear equations

determining the weights, it is clear that for when the maturity hedged corresponds to

one maturity in the hedging portfolio the weight for this maturity has to be one (up

to the discount factor), the others zero This last remark completely explains the

complex pattern we see.

- The extremely high values for the weights of the three factor model when the

hedging maturities are close – for example figure 4.5a where they are 2,3,4 months-

corresponds to a very low determinant for the matrix determining the weights. It

seems that, at least for this choice of parameters, the system is not well determined,

and quite unstable. For the three factor model, the sum of the weights decreases

very slowly toward 0 when the hedged maturity is long13, as shown on figure 4.12.

This visual analysis suggests, first, that using maturities too close on the hedge portfolio

could be a poor choice, and secondly that the three factor model could produce volatile

hedges. We will now discuss the numerical results of the hedge to see if it confirms our

initial impressions.

13 In fact provided the interest rates do not increase faster than maturities –in particular if they stabilize-, we see from the equations defining them that the hedging weights tend toward 0 when the hedged maturity tends toward infinity.

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Figure 4.12. Sum of the weights for the three-factor model when hedging a 5 years commitment with maturities of 2,6,12 months.

4.3.2.2. The efficiency of hedging strategies

In order to relate the hedge to the oil price we have chosen to give the values per barrel of

physical commitment.

Table 4.7. shows that the mean hedging errors for the hedging portfolios are not really

good, except for the 5 years commitment hedged with the three factor model of 2,6,9 months, or

2,6,12 months – which is in fact not really significant because we have only one replication of

this hedge. The hedge costs more than 5 $ per barrel. also notice an unexpected large gain with

the three factor model for the 2,3,4 months hedge, but again this is not significant due to the

small number of samples we have, and furthermore it is extremely volatile We A first

explanation of this result can be found in the poor performance of the models fitted on all the

data (from 1997 to 2002). A second – and more important - explanation will be given a bit later.

Looking more carefully at the table we see that the one-factor model perform reasonably well,

but that the two-factor model is the best on average.

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Table 4.7 Efficiency of the hedging strategies, Table 4.7 Efficiency of the hedging strategies, 5 Years 3 Years 1.5 Years 1 Year

Mean Vol Min Max Mean Vol Min Max Mean Vol Min Max Mean Vol Min Max1Fac M 2 -13,81 0,62 -1,84 1,10 -13,01 0,68 -2,69 1,44 -6,30 0,56 -2,69 1,38 -4,05 0,48 -2,69 1,301Fac M 6 -14,75 0,53 -1,72 0,98 -12,14 0,57 -3,47 1,33 -6,07 0,44 -3,47 1,25 -3,92 0,39 -3,47 1,251Fac M12 -15,99 0,46 -1,54 0,87 -10,36 0,60 -6,46 2,20 -5,50 0,64 -6,46 3,85 0,00 0,00 0,00 0,002Fac M 2,3 -10,88 1,07 -3,78 2,42 -10,23 0,96 -3,78 2,42 -5,57 0,74 -3,55 1,98 -3,72 0,54 -2,69 1,182Fac M 2,6 -11,76 0,60 -1,91 1,00 -10,78 0,55 -2,69 1,24 -5,74 0,40 -2,69 0,63 -3,80 0,31 -2,69 0,632Fac M 2,12 -12,13 0,44 -1,38 0,81 -11,06 0,41 -2,69 1,03 -5,80 0,31 -2,69 0,64 nc nc nc nc3Fac M 2,3,4 112,28 48,60 -152,1 108,85 195,35 36,98 -152,1 107,55 21,61 22,62 -152,1 74,89 nc nc nc nc3Fac M 2,6,9 -1,52 2,06 -4,00 7,25 -16,03 2,15 -6,88 11,29 -9,04 2,27 -6,88 11,29 nc nc nc nc3Fac M 2,6,12 -2,73 1,98 -3,56 7,25 -18,51 2,17 -6,86 11,21 -9,36 2,35 -6,86 11,21 nc nc nc nc

Going further in the analysis, we note a greater variance of the hedging errors, and a

greater error for hedges during the period of 1999. This corresponds to a change of regime of

the futures prices –see figure 4.1-. For example, the mean error for the hedge corresponding to

the 18 months commitment is less than 3. 10-2 for the hedges started early 1997, and reaches –

0.35 for the hedges started 15 to 20 months after. This is illustrated clearly in Figure 4.13 where

the monthly hedging errors for the 18 months commitment hedged with the two-factor model

and maturities of 2 and 12 months is shown in and while the red line corresponds to the same

hedge started 20 months later. These results seems to indicate that we could gain in performance

using a locally fitted model. The best would be to fit it on a moving window several months

long. On the other hand, we have also to remember that the models have been fitted on

maturities of 1,3,6,9 months. These maturities are suited for the optimisation of the parameters

because they ensure the stabilization of the optimization procedure. However, they are maybe

not exactly suited for all the hedging strategies chosen.

Figures 4.13. Hedging Errors when hedging a 18 Months commitment with the two factor model and with maturities of 2 and 12 months. The blue curve is the monthly error when startting the Hedge in 1997, the red

one when starting it mid-99.

The second explanation of our poor results lies in a practical problem with the strategy adopted. This problem arises when the larger maturity of the hedging portfolio is reached by the hedged commitment. As indicated previously, we have decided not to change the hedge portfolio. That is, to give an example, if we are hedging with the two-factor model, with futures

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of 2 and 12 months maturity, when the maturity of the hedged forward commitment falls below 12 months, we continue to hedge it with the same maturities. This has been done for comparison purposes. However, except for comparison concerns it is probably not the strategy that has to be recommended for real hedging. Indeed, when the maturity of the hedged forward commitment reaches the maturity of the futures positions, the better choice probably consists in abandoning the dynamic strategy and going back to traditional hedging (with one single position in the futures market).

of 2 and 12 months maturity, when the maturity of the hedged forward commitment falls below 12 months, we continue to hedge it with the same maturities. This has been done for comparison purposes. However, except for comparison concerns it is probably not the strategy that has to be recommended for real hedging. Indeed, when the maturity of the hedged forward commitment reaches the maturity of the futures positions, the better choice probably consists in abandoning the dynamic strategy and going back to traditional hedging (with one single position in the futures market).

Table 4.8 presents the results that we obtain with this most natural strategy, which

consists to stop to roll the position as soon we reach the maturity corresponding to the longest

dated future of our hedge portfolio. From this maturity to the maturity of the physical

commitment we will hedge using a future with the same maturity as the commitment - which

will not generate rolling errors-.

Table 4.8 presents the results that we obtain with this most natural strategy, which

consists to stop to roll the position as soon we reach the maturity corresponding to the longest

dated future of our hedge portfolio. From this maturity to the maturity of the physical

commitment we will hedge using a future with the same maturity as the commitment - which

will not generate rolling errors-.

Table 4.8 Efficiency of the hedging strategies with an adaptive hedge portfolio. Table 4.8 Efficiency of the hedging strategies with an adaptive hedge portfolio.

5 Years 3 Years 1.5 Years 1 YearMean Vol Min Max Mean Vol Min Max Mean Vol Min Max Mean Vol Min Max

1Fac M 2 -13,89 0,63 -1,84 1,10 -12,40 0,67 -1,85 1,44 -5,86 0,55 -1,70 1,38 -3,69 0,47 -1,70 1,301Fac M 6 -14,84 0,55 -1,72 0,98 -9,16 0,54 -1,72 1,33 -3,99 0,36 -1,49 1,07 -2,13 0,23 -1,27 0,811Fac M12 -11,38 0,49 -1,54 0,87 -4,86 0,41 -1,54 1,17 -1,64 0,18 -1,34 0,58 NC NC NC NC2Fac M 2,3 -11,26 1,09 -3,78 2,42 -9,04 0,97 -3,78 2,42 -4,73 0,75 -3,55 1,98 -3,03 0,55 -2,42 1,182Fac M 2,6 -12,10 0,62 -1,91 1,00 -7,67 0,54 -1,91 1,24 -3,63 0,36 -1,74 0,55 -2,01 0,22 -1,16 0,282Fac M 2,12 -12,13 0,44 -1,38 0,81 -4,53 0,33 -1,38 1,03 -1,55 0,18 -1,22 0,18 NC NC NC NC3Fac M 2,3,4 112,99 49,90 -152,1 108,85 196,38 38,64 -152,1 107,55 23,29 24,80 -152,1 74,89 NC NC NC NC3Fac M 2,6,9 -12,98 1,67 -4,00 3,26 -10,13 1,61 -5,31 7,67 -4,12 1,61 -5,31 7,67 NC NC NC NC3Fac M 2,6,12 -12,88 1,48 -3,56 4,30 -8,67 1,46 -4,43 6,20 -2,60 1,47 -4,43 6,20 NC NC NC NC

We see that this hedge performs far better. It is also clear from this table that strategies

with a larger maturity are on average better. This can be understood from an intuitive point of

view. It is better to hedge long term commitments with a future whose maturity is as close as

possible, in term of volatility and mean values they also compare better than small maturities

with long dated commitment. Finally, from this table it looks realistic to undertake an hedging

strategy up for commitments up to 3 years provided we have an enough liquid market up to one

year maturity. The one year hedge has been included for comparison with the work by Brennan

& Crew. Although the period of time and the length of the time series are different we find

comparable results.

We see that this hedge performs far better. It is also clear from this table that strategies

with a larger maturity are on average better. This can be understood from an intuitive point of

view. It is better to hedge long term commitments with a future whose maturity is as close as

possible, in term of volatility and mean values they also compare better than small maturities

with long dated commitment. Finally, from this table it looks realistic to undertake an hedging

strategy up for commitments up to 3 years provided we have an enough liquid market up to one

year maturity. The one year hedge has been included for comparison with the work by Brennan

& Crew. Although the period of time and the length of the time series are different we find

comparable results.

Finally, an attempt as been done using a different fitting of the parameters for the

model. We did not implement a moving window fitting, but we split the period of study in three

parts and fitted the parameters in each. Then, for each maturity the hedge was computed using

Finally, an attempt as been done using a different fitting of the parameters for the

model. We did not implement a moving window fitting, but we split the period of study in three

parts and fitted the parameters in each. Then, for each maturity the hedge was computed using

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the weights corresponding to the model valid for the corresponding period in time. Although the

performance in term of RMSE were better, the improvement for the hedging errors were poor.

the weights corresponding to the model valid for the corresponding period in time. Although the

performance in term of RMSE were better, the improvement for the hedging errors were poor.

4.3.2.3. Introducing the transaction costs : an example 4.3.2.3. Introducing the transaction costs : an example

In this work we have adopted a simple model for transaction costs. On the futures

market the transaction costs are by contract buy or sold. The range for professional is between

0.09 US$ to 0.20US$ for a contract, that is negligible for a barrel of oil compared to the hedging

error. So in this example we will neglect these costs. On the interest rate market, we model them

as a spread. We will show the results for a spread of 40 BP (basis point) and another for 100BP.

We choose not to modify the hedging weights according to the value of spread. In the examples

the spreads affects only the present value of the physical commitment rolled each months (the

Lt,ν terms) and the monthly benefit of our loan. Table 4.9 presents the case of a 40 BP spread

and 4.10 the case of a 100BP spread. All the value are in percent relative to the hedge without

spread.

In this work we have adopted a simple model for transaction costs. On the futures

market the transaction costs are by contract buy or sold. The range for professional is between

0.09 US$ to 0.20US$ for a contract, that is negligible for a barrel of oil compared to the hedging

error. So in this example we will neglect these costs. On the interest rate market, we model them

as a spread. We will show the results for a spread of 40 BP (basis point) and another for 100BP.

We choose not to modify the hedging weights according to the value of spread. In the examples

the spreads affects only the present value of the physical commitment rolled each months (the

Lt,ν terms) and the monthly benefit of our loan. Table 4.9 presents the case of a 40 BP spread

and 4.10 the case of a 100BP spread. All the value are in percent relative to the hedge without

spread.

Table 4.9 Effect on the hedge of an interest rate spread of 40 BP Table 4.9 Effect on the hedge of an interest rate spread of 40 BP

5 Years 3 Years1Fac M 2 1,53% 1,15%1Fac M 6 1,46% 1,23%1Fac M12 1,42% 1,39%2Fac M 2,3 1,73% 1,37%2Fac M 2,6 1,67% 1,33%2Fac M 2,12 1,66% 1,31%

Table 4.10 Effect on the hedge of an interest rate spread of 100 BP Table 4.10 Effect on the hedge of an interest rate spread of 100 BP

5 Years 3 Years1Fac M 6,21% 4,66%1Fac M 5,95% 4,96%1Fac M1 5,77% 5,62%2Fac M 7,02% 5,54%2Fac M 6,78% 5,37%2Fac M 6,76% 5,29%

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4.4. Conclusion of the fourth part

The procedure for applying the general methodology presented in the previous parts can

be split into several steps.

The first one is simply the model calibration. We chose to calibrate the three models by

combining maximum likelihood and Kalman filters. One of the main practical difficulties is

encountered when trying to maximize the likelihood under constraints. The objective function is

far from simple with many parameters to be estimated. Furthermore for the three factor model

we have non-linear constraints for the correlation coefficients. The experimental results

confirmed our initial impression of the three factor model: although there are more degrees of

freedom the gain is negligible. This is in part because the third factor has roughly the same

behaviour as the second.

The second step consists in implementing the hedging methodology. To have a better

physical feeling for the results we examined the weights produced by each of the three models.

Two facts appear:

• The three factor model can produce larger weights.

• The initial strategy to pursue the hedge by rolling a predefined portfolio until maturity

leads to strong variations in the weights when the residual maturity is shorter than the

highest maturity of the hedge portfolio.

This is confirmed by the experimental results obtained on the 1997-2002 Nymex data. The

results for the 5 year commitment are not reliable because we have only one replication of the

strategy. But for the other contracts investigated we have a more reasonable sampling. The

dynamic hedge appears to be efficient up to about 18 months to 3 years, provided we stop

rolling the hedge as soon we reach the higher maturity of the hedge portfolio. As expected

hedging with a large maturity future is more efficient. But the practical limitation is the liquidity

of the future contracts. This is the reason why we limit the maturity of the future portfolio to 12

months .

We also investigated the effects of rolling the portfolio every two or three months instead

of each month. In our case this had a small impact. Finally the effect of the transaction costs has

been illustrated on one example considering only the effect of an interest rate spread, which

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starts to have a practical impact once the spread is around 100 BP.

Some care is required when interpreting the results because the calibration was carried out

using all the data (rather than moving windows) and also because it was based on the

experimental errors, via the likelihood. Because the period considered is large this result on root

mean squared errors is not negligible. But a calibration in 3 time periods did not lead to a

significant improvement, even if the RMSE was about 20 to 30% better.

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Appendix 4. Kalman filter and parameters’ estimation

The Kalman filters are powerful tools, which can be used for models estimation in many areas in finance. A Kalman filter is an interesting method when a large volume of information must be taken into account, because it is very fast. When associated with an optimization procedure, it can also be used for the estimation of the parameters, if the model relies on non-observable data.

The simple Kalman filter is the most common version of the Kalman filter. It can be used when the model is linear, as is the case for Schwartz’ model. First, the main principles of the method are presented. Second, the parameters estimation is exposed.

1. Presentation14

The main principle of the Kalman filter is to use temporal series of observable variables in order to reconstitute the value of non-observable variables. The model has to be expressed in a state-space form characterized by a transition equation and a measurement equation. Once this has been made, a three step iteration process can begin.

The state-space form model, in the simple filter, is characterized by the following equations:

• Transition equation:

tttt RcT ηαα ++= −− 11/ (5)

where αt is the m-dimensional vector of non-observable variables at t, also called state vector, T is a matrix (m × m), c is an m-dimensional vector, and R is (m × m)

• Measurement equation:

ttttt dZy εα ++= −− 1/1/ (6)

where is an N-dimensional temporal series, Z is a (N×m) matrix, and d is an N-dimensional vector.

1/ −tty

tη and tε are white noises, which dimensions are respectively m and N. They are supposed to be normally distributed, with zero mean and with Q and H as covariance matrices:

14 Harvey (1989) inspires this presentation.

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[ ] 0=tE η and [ ] QtVar =η

[ ] 0=tE ε and [ ] HtVar =ε

The initial value of the system is supposed to be normal, with mean and variance:

[ ] 00~αα =E , [ ] 00 PVar =α

If tα~ is a non biased estimator of αt, conditionally to the information available in t, then:

[ ] 0~ =− tttE αα

Consequently, the following expression15 defines the covariance matrix Pt :

( )( )[ ]'~~tttttt EP αααα −−=

During the iteration, three steps are successively tackled: prediction, innovation and

updating.

• Prediction:

+=+=

−−

−−

''

~~

11/

11/

RQRTPTPcT

ttt

ttt αα (7)

where 1/~

−ttα and Pt / t-1 are the best estimators of αt/t-1 and Pt/t-1 , conditionally to the information available at (t-1).

• Innovation:

+=−=

+=

−

−

−−

HZZPFyyv

dZy

ttt

tttt

tttt

'

~~~

1/

1/

1/1/ α (8)

where is the estimator of the observation y1/~

−tty t conditionally to the information available at (t-1), and vt is the innovation process, with Ft as a covariance matrix.

• Updating:

−=

+=

−−

−

−−−

1/1

1/

11/1/

)'(

'~~

tttttt

ttttttt

PZFZPIP

vFZPαα (9)

The matrices T, c, R, Z, d, Q, and H are the system matrices associated with the state-space model. They are not time dependent in the case considered in this report.

)15 ( '~

tt αα − is the transposed matrix of ( )tt αα −~ .

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2. Parameters’ estimation

When the Kalman filter is applied to term structure models of commodity prices, the aim is the estimation of the parameters of the measurement equation, in order to obtain estimated futures prices for different maturities ( )iF τ~ , and to compare them with empirical futures prices ( )iF τ . The closest the firsts are with the seconds, the best is the model.

Suppose that the non-observable variables and the errors are normally distributed. Then the Kalman filter can be used to estimate the model parameters, which are supposed to be constant. On that purpose, the logarithm of the likelihood function is computed for the innovation vt, for given iteration and parameters vector:

tttt vFvdFntl ××−−Π×

−= −1'

21)ln(

21)2ln(

2)(log (10)

where Ft is the covariance matrix associated with the innovation vt, and dFt its determinant16.

Relying on the hypothesis that the model measurement equation admits continuous partial derivatives of first and second order on the parameters, another recursive procedure is used to estimate the parameters17. An initial M-dimensional vector of parameters is first used to compute the innovations and the logarithms of the likelihood function. Then the iterative procedure researches the parameters vector that maximizes the likelihood function and minimizes the innovations. Once this optimal vector is obtained, the Kalman filter is used, for the last time, to reconstitute the non-observable variables and the measure y~ associated with these parameters.

3. Application to the one-factor model

The simple Kalman filter can be applied to the one-factor model because the later can be easily expressed on a linear form, as follows:

( ) ( ) ( )ttt eeSeTtSF κκκ

κσα 2

2

14

1ˆln),,(ln −−− −+−+=

The transition equation is inspired by the dynamic of the state variable:

16 The value of logl(t) is corrected when dFt is equal to zero.

17 The optimization algorithm used is the one proposed by Broyden, Fletcher, Goldfarb and Shanno

(BFGS).

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with : - X = ln S and κ

σµα2

2

−=

The transition equation is:

tttt XTcX η+×+= −− 11/~~ , t = 1, ... NT (13)

where:

- tc ∆= κα , and ∆t is the period separating 2 observation dates

- tT ∆−= κ1

- ηt are uncorrelated errors, with :

E[ηt] = 0, and tVarQ St ∆== 2][ ση

The measurement equation is:

ttttt XZdy ε+×+= −− 1/1/~~ , t = 1, ... NT (14)

where:

- the ith line of the N dimensional vector of the observable variables is 1/~

−tty ( )( )[ ]iF τ~ln , with

i = 1,..,N, where N is the number of maturities retained for the estimation.

- ( ) (

−+−= −− ii eed κτκτ

κσα 2

2

14

1ˆ ) is the ith line of the d vector, with i = 1,..., N

- [ ]ieZ κτ−= is the ith line of the Z vector, with i = 1,...,N

- εt is a white noise’s vector, (N×1), with no serial correlation : E[εt] = 0 and H = Var[εt]. H

is (N × N)

4. Application to the two-factor model

The simple Kalman filter can be applied to the two-factor model because the later can be easily expressed on a linear form, as follows:

( ) ( ) ( )τκ

κτ

BetCtSTtCSF +−

×−=−1)()(ln),,,(ln (11)

Letting G = ln(S), we also have:

[ ]

21( )2 S S

C C

dG C dt dz

dC kC dt dz

= µ − − σ + σ = − + σ

S (12)

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The transition equation is:

tt

t

tt

tt RCGTc

CG

η+

×+=

−

−

−

−

1

1

1/

1/ ~~

~~

, t = 1, ... NT (13)

where:

- 21

20

S tc

µ − σ ∆ =

is a (2×1) vector, and ∆t is the period separating 2 observation dates

- is a (2 × 2) matrix,

∆−

∆−=

tt

Tκ10

1

- R is the identity matrix, (2 × 2),

- ηt are uncorrelated errors, with :

E[ηt] = 0, and

∆∆∆∆

==tt

ttVarQ

CCS

CSSt 2

2

][σσρσ

σρσση

The measurement equation is:

ttt

tttt C

GZdy ε+

×+=

−

−−

1/

1/1/ ~

~~ , t = 1, ... NT (14)

where:

- the ith line of the N dimensional vector of the observable variables is 1/~

−tty ( )( )[ ]iF τ~ln , with

i = 1,..,N, where N is the number of maturities retained for the estimation.

- d = [B(τi)] is the ith line of the d vector, with i = 1,..., N

-

−−=

−

κ

κτ ieZ 1 ,1 is the ith line of the Z matrix, which is (N×2), with i = 1,...,N

- εt is a white noise’s vector, (N×1), with no serial correlation : E[εt] = 0 and H = Var[εt]. H

is (N × N)

5. Application to the three-factor model

Let ξt be the vector whose ith component is =ln(F(t,Ti ,S,y,ν), so that we can write:

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, ln( ) ( , ) ( , ) ( )i t t t i t i iS y H k H aξ = − τ + ν τ + ϕ τ

if we write gt=ln(St), we see that ξi,t is a linear combination of a Gaussian.

Now letting:

t

t t

t

gyα =ν

we can write the transition equation, after discretization using the Euler scheme as:

t t 1T c R− tα = α + + η

where :

1 t tT 0 1 k t 0

0 0 1 a

−∆ ∆= − ∆

t− ∆

212

c 00

σ

t

+ µ

= ∆

( )

12 12

23 12 13 23 12 1313 13

1212

12

2 2

22

3 3 3 22

0 0

R 1 0

111

σ

= σ ρ σ − ρ ∆

ρ − ρ ρ ρ − ρ ρσ ρ σ σ − ρ −

− ρ− ρ

t

and is a vector of 3 mutually independent gaussian variable with zero mean and variance. tη

Note that in the expression given for R it is assumed that:

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2 223 12 13 12 13( ) (1 ) (1ρ − ρ ρ ≤ − ρ − ρ 2)

t

, which will give an additional –non linear –

constraint for the maximisation of likelihood.

The measurement equation can be written in the following way:

t tZ dξ = α + + ε

where :

( , ) ( , )( , ) ( , )

. . .( , ) ( , )

1 1

2 2

n n

1 H k H a1 H k H a

Z

1 H k H a

− τ τ− τ τ

=

− τ τ

( )( ).

( )

1

2

n

d

ϕ τϕ τ

=

ϕ τ

Where the εt is a normal vector not serially correlated, whose components are not correlated.

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Conclusion of the report

In this report we have investigated the experimental performance of a dynamic hedging

strategy to secure a physical commitment having a one to five years maturity. In order to

implement this strategy a model for the term structure is needed. We selected three models from

those available: a simple one-factor model, where the single factor, the spot price, is assumed to

be the exponential of an mean reverting process (namely an Ornstein Uhlenbeck process), a

two- and a three-factor models, where the spot price is a geometric brownian motion, with a

drift defined by one or two Ornstein Uhlenbeck processes.

The parameters of these models have been calibrated applying a Kalman filter and

maximum likelihood to the data, which consist in the values of the crude oil futures prices up to

5 years, for the period 1997-2002. The one, three, six and nine months maturities have been

used for this calibration. The others maturities have been used in order to experimentally test the

performances of the dynamic hedging strategies. In addition to the futures prices values, we

took into account the effect of the term structure of interest rates, which was available to us for

the same period than the futures.

The experimental results show that although the model has been fitted on the whole

period (1997-2002), the dynamic hedging can be efficient with a hedge portfolio incorporating

maturities less than 12 months, provided we limit ourselves to 1.5 years to 3 years ahead. The

results we obtain are comparable to those obtained by Brennan & Crew on a different period in

time and considering a one and two factor model, but with a constant interest rate. In our case

the effect of transaction costs on the futures market are negligible, but the interest rate spread on

the hedger has an impact (of several percent) on the hedge performance.

To improve the hedging performance we have to improve the adequacy of the model to

our data. This can be done by changing the way we calibrate the models. This is the main

direction for our future research.

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Documents

Dynamic hedging strategies and commodity risk … hedging strategies and commodity risk management D. Lautier & A. Galli Mercantile Exchange, and American interest rates corresponding