Hedging prices of oil products: a comparison
between mutual and third-party strategies
Guido Arons*
5700817
Abstract
This thesis describes how mutual and third-party hedging strategies can be compared with each other. Kerosene-expenses are hedged mutually with gas sales. Moreover third-party hedge strategies are implemented that hedge kerosene with kerosene derivatives and gas with gas options. From the considered hedge strategies it is found that both rising third-party fees and an increasing price correlation between mutually hedged products make mutual hedging strategies relatively more attractive than third-party strategies. The marginal effects of an increasing correlation and increasing fees are both decreasing. The attractiveness of a strategy is expressed in terms of the level of risk tolerance at which agents are indifferent between mutual and third-party contracts. A decrease in risk tolerance makes third-party hedging strategies more attractive than hedging mutually. Using mutual options that are developed by taking into account that the ratio between kerosene and gas prices can vary, instead of mutual options whose strike are chosen assuming that the price ratio is fixed, makes mutual hedging relatively more attractive.
* Master’s Thesis Econometrics
University of Amsterdam, Department of Economics
December 2007
EMBARO FOR RELEASE UNTIL JANUARY 2011 EXCEPTING SUPERVISOR AND SECOND EXAMINER
Supervisor: Prof. Dr. H.P. Boswijk Second examiner: Dr. M.J.G. Bun
Preface
The attention on financial risk management has increased over the last years. Companies realize that, beside the impact of their own decisions, external factors like commodity prices can affect their performance heavily. Shareholders can expect the company to focus on its core business instead of gambling on the commodity markets since shareholders themselves can take positions in commodity markets. Derivative strategies are a popular way to manage financial risks. Third parties like banks are well-known suppliers of derivatives. However, third parties are in general not willing to take over the risk of company. Instead they try to hedge these risks with contracts with parties that have opposite positions. By hedging mutually, parties with opposite positions on a commodity can choose to avoid costs charged by a third-party. This research explores how a comparison can be made between third-party hedge contracts on the one hand and mutual hedge contracts on price correlated products on the other hand. The considered products are kerosene and natural gas as the prices of these products are, among others, risk drivers of Air-France KLM, Royal Dutch Shell, Royal Vopak, and the Dutch Ministry of Economic Affairs. I would like to thank these parties for showing their interest in this potential way of hedging, facilitating in collecting the required data and helping me to understand their ways and purposes of financial risk management. Furthermore, I would like to thank Prof. Dr. H.P. Boswijk for the allowed freedom with respect to choices made in this research. Our discussions and your suggestions are a great contribution to the thesis.
Content overview
Chapter 1 Introduction 2
Chapter 2 Research setup 4
Chapter 3 Modelling prices 6
3.1 Objectives of price models 6
3.2 Selecting a method to models prices 8
3.3 A historical price model 12
3.4 An external factor model 16
3.5 A kerosene price model 21
Chapter 4 Option strategies 24
4.1 Mutual option strategies 25
4.2 Third-party option strategies 37
4.3 Parameter hedging 39
Chapter 5 Simulation and evaluation 46
5.1 Simulation 47
5.2 Evaluation 53
Chapter 6 Conclusions and recommendations for further research 57
Appendix A Unit root tests 58
Appendix B Test statistics for the historical price model 60
Appendix C Information on variables 61
Appendix D Test statistics for the external factor model 62
Appendix E Test statistics for the kerosene price model 63
Appendix F Strikes and payment schedules for mutual options at original correlation 64 Appendix G Strikes and payment schedules for mutual options at decreased correlation 67 Appendix H Strikes and payment schedules for mutual options at increased correlation 70 Appendix I Strikes and payment schedules for third-party options at original correlation 73 Appendix J Strikes and payment schedules for third-party options at decreased correlation 76 Appendix K Strikes and payment schedules for third-party options at increased correlation 79
1 Introduction
For economic agents who buy or sell products over a long time horizon the financial risks related to the price of the product are of major importance for the continuity of their activities. Agents can choose which price risks they are willing to take and which risks they want to avoid. One way to avoid certain price risks is to take positions in the option market of the specific product. Intermediary parties are often willing to offer the requested option and will in general charge a transaction fee, margin and risk premium. Lowering the price risk by buying options will, because of these additional costs, consequently lower the agent’s expected profit. If two agents face different risks related to the price of the same product the risk exposure of the parties can be opposite with respect to their profits: for all realisations of the price of the product that differ from its expected value, one of the two agents receives a higher than expected profit where the other has a lower than expected profit. If one of these agents sells a specified option to the other, the risks of both parties can be reduced. The advantage of this mutual transaction is the elimination of transaction fees and margins. A mutual option transaction can also take place between agents who face opposite risks related to the price of positively correlated products. However, although prices are correlated there may occur situations where both agents face lower than expected profits as a consequence of a decreased price of one product and an increased price of the other product. These situations cannot occur when agents with opposite positions on the price of the same product set up an option contract. The last example exposes the disadvantage of an option between parties with opposite positions on correlated products compared to third-party contracts on a single product.
The subject of mutual hedging is relatively unexplored in literature. Dowd (2003) states that parties with an opposite mortality risk exposure can consider exploiting their mutual hedging potential. A comparison between mutual and third-party contracts is not made and the aspect of correlation between underlying products is not applicable in his article. Recently Spargoli and Zagaglia (2007) researched the co-movements between future markets for crude oil in New York and London. They find evidence that futures on both markets can be hedged successfully with each other as long as turbulent market circumstances do not occur. The possibility to hedge mutually is shown in this article, but again a comparison between mutual and third-party contracts is not made. The question arises whether a mutual hedge, with an instrument that has a product as underlying whose price is correlated to the product that one wishes to hedge, is a better choice than a third-party hedge with an instrument that has the hedged product as underlying.
This thesis explores how the choice between mutual and third-party option contracts is affected by risk tolerance, price correlation and third-party fees and margins. The research question is formulated in the following way: How does the rational choice between mutual option contracts on price-correlated products and third-party contracts on a single product depend on fees, risk tolerance and price correlation? As mentioned in the preface, the products for which option contracts are considered are kerosene and natural gas.
In order to answer the research question several steps with different econometric methods will be made. The first step consists of developing a model that defines the relationship between the prices of kerosene and natural gas. Two models are formulated which allows comparing the robustness of the models on which option contracts are based. The first model is an ARMA-GARCH model, which gives a forecast of the future kerosene-gas-price ratio by using historical price ratios only. The second model uses a completely different concept: supply and demand related factors of both products together with other external factors provide a forecast of the future price ratio. Not only the price ratio but also the absolute prices are requested to close an option contract. Therefore, a model is estimated that explains the price of kerosene by using the historical prices. Together with the price ratio the price of natural gas can then be obtained. The second step consists of the development and performance evaluation of option strategies. Price and parameter hedge strategies are designed using mutual and third-party contracts. Price hedge strategies offer direct protection to changes of prices whereas a parameter hedge reduces the exposure of agents to changing parameter values of the price generating model. In order to evaluate the performance of the option contacts, the contracts are valued for simulated price and parameter scenarios. The robustness of the price ratio models is observed in a cross model comparison: an option strategy designed for one price ratio model is valued for price scenarios simulated with the other price ratio model and vice versa. Finally a risk-aversion analysis is made which allows comparing the performances of the option strategies and the price ratio models.
The remaining part of the thesis is organised in the following way. Conceptual information on the setup of the research is provided in chapter two. Chapter three is dedicated to the estimation of the price ratio models and the kerosene price model. In chapter four, hedge strategies are developed. The succeeding chapter five describes how prices are simulated in order to value the option contracts. The results found are also evaluated in this chapter. Finally the conclusions of the thesis are drawn in chapter six.
2 Research setup
In order to answer the research question formulated in the introduction a number of necessary steps is executed in this thesis. This chapter provides information on what steps are implemented and how they are related to each other.
Economic agents in this research are assumed to make a rational choice between mutual option contracts on price-correlated products and third-party contracts on a single product. A utility function on the mean and variance of the net pay-off of an option strategy will be used to compare both types of strategies. This utility function contains a risk tolerance parameter on which assumptions will be made. To determine the mean and the variance of the net pay-off of an option strategy the cost of options is a required factor. Furthermore a price simulation provides information on the mean and the variance of the pay-off of the option strategy. Both the cost and the pay-off of an option depend on its strike and information on the distribution of the future price. This distribution can be derived from models that describe the relation between kerosene and gas, which will be estimated in the thesis. These models will also be used for the price simulation. For the cost of third-party options, fees are an additional determinant on which assumptions will be made in this research.
The choice for the strike of an option depends on the risk management policy of the agent and information on the distribution of the future price: which price deviations are acceptable and against which deviations does the agent wish to be protected? The information on the distribution of the future price is again derived from the estimated price models. In addition assumptions on the risk management policy of the agents will be made. Finally, the impact of changes of the correlation between kerosene and gas prices on the choice between mutual and third-party contracts will be researched. This will be accomplished by modifying the estimated relation between kerosene and gas prices and observing the effects on the rational choice between mutual and third-party contracts. The above-described steps are graphically presented in schedule 2.1 on the next page.
Schedule 2.1: Concept of the research
The first step in answering the research question is estimating the price models, which is executed in the next chapter. In chapter four the strikes and costs of the options are developed. Assumptions on the risk management policy and third-party fees are made in chapter four as well as modifications of the estimated relation between kerosene and gas prices. The pay-off of options is simulated in chapter five. This chapter also contains assumptions on risk tolerance of the agents and an evaluation of the simulations results. With the information found in chapter five conclusions can be drawn, which is performed in chapter six.
price models +
modifications of relation between kerosene and gas price
assumptions on risk management policy strike of options assumptions on third party fees price simulation engine cost of options assumptions on risk tolerance pay-off of options comparison between option strategies
3 Modelling prices
One of the major causes of financial risks is price uncertainty. Option structures can be implemented to decrease the impact of changes of prices on profits. This research compares mutual option contracts on kerosene and gas prices to third-party contracts. Therefore, not only the prices of kerosene and gas are relevant; the relation between the prices of these fuels is important to consider as well. In this chapter two models are developed that describe this relation. An additional model provides a forecast of the future kerosene price. Together with the models that describe the relation with the gas price, the price of gas can be obtained. In advance of the development of the three models the next two sections provide information on the objectives and methods of the models.
3.1 Objectives of price models
As mentioned in the introduction of this chapter, option structures can be used to decrease the financial risks caused by price uncertainty. If an agent does not have any knowledge on the development of prices that are relevant for his portfolio, a price hedge is the most obvious way to reduce risks. Put and call options that have the agent’s product as underlying can then guarantee the future turnover or cost to be in a certain range.
However, the prices of products that are relevant for the participating parties of this research, natural gas and kerosene, are supposed to be related to each other. In order to design a mutual option contract on correlated products, it is necessary to estimate a model that describes the relation between the prices of the products. This model provides information on the confidence intervals of the future price. By the definition of a confidence interval, prices can occur that lay outside the confidence intervals. Furthermore, the financial performance of an agent may also suffer from unfavourable prices that are in the interval. Therefore, even if the agent has information on confidence intervals of future prices it is recommended from the perspective of risk management to maintain a price hedge.
The extra information that a price model provides is useful to extend this price hedge with a more advanced hedge type. Economic processes develop over time; ignoring this development will cause an underperformance of a price model and simultaneously higher price uncertainty.
Translated in terms of a model, changing characteristics of an economy can result in modified parameter estimates. The effect of a parameter change of the price model can be reduced with a so-called parameter hedge. One has to estimate a price model in order to determine how changing parameter values affect the future profit of an agent. In this chapter two models are estimated that describe the relation between gas and kerosene prices. The goal of the option strategy, which is formulated later on in chapter four, is to decrease price risks over a single period ahead. Strategies that decrease risks over more than one period could also have been developed. However, the purpose of this research is to compare third-party contracts to mutual options; a comparison of the effectiveness of hedges of different period lengths is not in the scope of this research. Therefore, the developed models forecast the relation between kerosene and natural gas prices one period ahead. The chosen period length is one month, because a substantial part of the explanatory variables is available on a monthly base only.
The monthly average price of gas is computed from its daily last prices of one-day-ahead delivery terms of gas at the National Balancing Point in the UK. This is both the oldest and most liquid European market. Kerosene prices are observed in US Dollars per barrel in Rotterdam, since this is the most liquid European market as well. As mutual contracts on price-correlated products and not currency risks are central in this thesis, the UK gas prices are converted from British Pounds into US Dollars by using daily exchange rates. The National Balancing Point gas market was introduced in April 1996; the data are collected from then until February 2007.
3.2 Selecting a method to model prices
There are several methods to describe the relation between gas and kerosene prices. In this section the time series of kerosene, natural gas and the ratio of kerosene and gas prices are studied with respect to their stationarity. The existence of unit roots for the different time series is tested by augmented Dickey-Fuller tests, introduced by Dickey and Fuller (1979). With this information a method can be selected to model the relation between the prices of both fuels.
The monthly average kerosene price is plotted in figure 3.2.1 .
Figure 3.2.1: Monthly average kerosene price
Augmented Dickey-Fuller tests are performed on both the series of the price and the natural logarithm of the price. These tests are performed without an intercept and a trend, with an intercept, and with both an intercept and a trend in the test equation. Based on the one-sided critical values introduced by Mackinnon (1996) the hypothesis that the series has a unit root could not be rejected using tests up to twelve lags for both the kerosene price and the log kerosene price. The p values of these tests can be found in appendix A.
Monthly average kerosene price
0 10 20 30 40 50 60 70 80 90 100 1 9 9 6 m 0 4 1 9 9 7 m 0 4 1 9 9 8 m 0 4 1 9 9 9 m 0 4 2 0 0 0 m 0 4 2 0 0 1 m 0 4 2 0 0 2 m 0 4 2 0 0 3 m 0 4 2 0 0 4 m 0 4 2 0 0 5 m 0 4 2 0 0 6 m 0 4 US Dollar / barrel t
In figure 3.2.2 the monthly average nbp gas price is graphically represented.
Figure 3.2.2: Monthly average nbp gas price
The same augmented Dickey-Fuller tests are performed on both the series of the price of gas and the natural logarithm of the price as performed on the kerosene price series. Appendix A contains the p values of the tests. As for the kerosene price series, the hypothesis that the series has a unit root could not be rejected using tests up to twelve lags for gas prices and log gas prices.
Figure 3.2.1 and 3.2.2 suggest that the ratio between kerosene and gas prices shows stationarity. This relation can be explained by noticing that, although kerosene and natural gas can in general not be regarded as substitutes for each other, their prices are both linked with the oil price and, therefore, with each other.
Monthly average nbp gas price
0 0.2 0.4 0.6 0.8 1 1.2 1.4 1 9 9 6 m 0 4 1 9 9 7 m 0 4 1 9 9 8 m 0 4 1 9 9 9 m 0 4 2 0 0 0 m 0 4 2 0 0 1 m 0 4 2 0 0 2 m 0 4 2 0 0 3 m 0 4 2 0 0 4 m 0 4 2 0 0 5 m 0 4 2 0 0 6 m 0 4 US Dollar / term t
The kerosene-gas-price ratio is plotted in figure 3.2.3.
Figure 3.2.3: Kerosene-gas-price ratio
On both the series of the ratio and the natural logarithm of the ratio the same augmented Dickey-Fuller tests are performed as done on the kerosene price and the gas price series. In appendix A the p values of the tests can be found. Opposite to the outcome for the kerosene and gas price series, the hypothesis that the series has a unit root could be rejected using tests with up to twelve lags if an intercept is included in the test equation. This result holds for both the ratio as the log ratio series. The fact that an intercept is needed can be clarified by observing that the mean of the kerosene-gas-price ratio is not zero. In section 3.3 a GARCH model is estimated to get insight in volatility clustering. As this model is stationary, the Dickey-Fuller test statistic is asymptotically distributed with the distribution under the null hypothesis.
For the purpose of designing mutual option contracts on correlated products, the ratio between both prices can be useful. This ratio clearly indicates which amount of terms of gas is expected to have the same value as one barrel of kerosene. VAR models with an extensive number of explanatory variables were also estimated. However, these models had less explanatory power in terms of R2 than the models that explain the kerosene-gas-price ratio and are, therefore, not included in the thesis.
Kerosene-gas-price ratio 0 50 100 150 200 250 1 9 9 6 m 0 4 1 9 9 7 m 0 4 1 9 9 8 m 0 4 1 9 9 9 m 0 4 2 0 0 0 m 0 4 2 0 0 1 m 0 4 2 0 0 2 m 0 4 2 0 0 3 m 0 4 2 0 0 4 m 0 4 2 0 0 5 m 0 4 2 0 0 6 m 0 4 kerosene/nbp t
As mentioned in the introduction of the thesis two models that describe the relation between kerosene and gas prices are formulated, which allows comparing the robustness of the models on which option contracts are based. In order to make a reasonable comparison between two models, a clear difference is made in their design. The first model, called historical price model, uses the historical price ratios only to explain the month-ahead price ratio. The second model, named external factor model, is built on supply, demand and other external factors. Finally a model is estimated that explains kerosene prices one month ahead. The information of this model, together with the price ratio models, also provides a one-month-ahead forecast of the price of natural gas.
Instead of the ratio of the kerosene price and the gas price the log of this ratio also could have been selected as endogenous variable. Log linear variants of the historical price model and the external factor model were estimated as well. However, these models had less explanatory power in terms of R2 than the models that included the pure ratio of kerosene and gas prices as endogenous variable and are, therefore, not included in the thesis.
3.3 A historical price model
The first model has an autoregressive moving average specification, abbreviated as ARMA (m,p). The ratio of kerosene and gas prices is explained by historical ratios. The model is specified in the
following way: kt m kt-i p = µk/nbp + Σ ai - µk/nbp + Σ bjεt-j + εt , εt ~ N(0, σ 2 t) (3.3.1) nbpt i=1 nbpt-i j=1 with
kt as the price of kerosene in month t;
nbpt as the price of National Balancing Point gas in month t;
µk/nbp as the mean of the kerosene-gas-price ratio;
m as the number of included autoregressive terms; p as the number of included moving average terms; ai as the coefficient of the i-th autoregressive term;
bj as the coefficient of the j-th moving average term;
εt as the error term in month t and
σ2
t as the estimated variance of the error term in month t.
As estimator for µk/nbp the sample mean of the kerosene-gas-price ratio is used. With respect to
the distribution of the error terms of (3.3.1) a model is estimated to explain the variance. The selected model is an autoregressive conditional heteroskedasticity model introduced by Engle (1982) and extended by Bollerslev (1986) to the generalized version by adding lagged conditional variances. This GARCH(q,r) model is defined in the following way:
q r σ2t = ω + Σ ciε 2 t-i + Σ djσ 2 t-j (3.3.2) i=1 j=1 with σ2
t as the estimated variance of the error term of (3.3.1) in month t;
ω as a constant;
εt as the error term of model component (3.3.1) in month t;
q as the number of included squared error terms; r as the number of included estimated variance terms; ci as the coefficient of the i-th squared error term and
dj as the coefficient of the j-th estimated variance term.
For simultaneously estimating the combined model of (3.3.1) and (3.3.2) the values of m, p, q and r should be chosen is such a way that the explanatory power of the model in terms of R2 is optimised under the restriction that the standard residuals do not show autocorrelation or heteroskedasticity. A model with chosen numbers for m, p, q, and r is declined if the hypothesis that the residuals are not autocorrelated is rejected. This hypothesis is tested by using the Ljung-Box Q-statistic at a 5% significance level. An ARCH-LM test is used to test the hypothesis that there is no additional autoregressive conditional heteroskedasticity. If this hypothesis is rejected at a 5% significance level the model is modified with respect to q and r. Since it is assumed that the influence of residuals of more than one year ago can be ignored, both tests are performed with twelve lags included. Furthermore, a model is reduced if one of the a, b, c or d type parameters is insignificant. This is tested by using a t-statistic at a 5% significance level. The numerical outcomes of this test and the other tests announced in this section can be found in appendix B.
Maximum Likelihood is used to estimate the models. The model is estimated under the assumption that the standard errors of the estimators are conditionally normally distributed. From the considered normality test based on the Jarque-Bera statistic the hypothesis that the residuals are conditionally normally distributed was not rejected. Therefore, using quasi maximum likelihood was redundant; maximum likelihood was used for the parameter estimation. The described estimation method leads to the following ARMA(1,2)-GARCH(1,1) model:
kt kt-1 = 126.469 + 0.480 - 126.469 + 0.311 εt-1 + 0.217 εt-2 + εt , εt ~ N(0, σ 2 t) nbpt (0.157) nbpt-1 (0.136) (0.105) (3.3.3) σ2t = 3.803 + 0.0714 ε t - 1 2 + 0.915 σ 2t - 1 (3.3.4) (1.318) (0.028) (0.413)
T-statistics of the estimated coefficients are provided in appendix B. Figure 3.3.1 provides a graphical representation of the actual values of the kerosene-gas-price ratio and the values fitted by model (3.3.3).
Figure 3.3.1: Actual and fitted values of the historical price model for the kerosene-gas-price ratio
The R-squared of the regression equals 0.784. By using the information of prices and error terms of January and February 2007, it is possible to make a forecast for the price ratio for the next month. Computing the expectation of kerosene-gas-price ratio for March 2007 leads to a value of 163.442. With respect to the variance of the forecast equation (3.3.4) is used. Since it holds in the estimated GARCH (1,1) model that c1 + d1 = 0.0714 + 0.915 = 0.989 < 1, the unconditional
variance exists and equals ω /(1- c1 - d1) = 3.803 /(1 – 0.0714 – 0.915) = 279.642. Equation
(3.3.4) also provides estimations of the variance of the kerosene-gas-price ratio over the observed period. The standard deviations derived from the estimated variances are presented in figure 3.3.2.
Ratio of kerosene and gas prices
0 50 100 150 200 250 1 9 9 6 m 0 4 1 9 9 7 m 0 4 1 9 9 8 m 0 4 1 9 9 9 m 0 4 2 0 0 0 m 0 4 2 0 0 1 m 0 4 2 0 0 2 m 0 4 2 0 0 3 m 0 4 2 0 0 4 m 0 4 2 0 0 5 m 0 4 2 0 0 6 m 0 4 t kerosene/nbp actual values model fitted values
Estim ated standard deviation 0 5 10 15 20 25 1 9 9 6 m 0 4 1 9 9 7 m 0 4 1 9 9 8 m 0 4 1 9 9 9 m 0 4 2 0 0 0 m 0 4 2 0 0 1 m 0 4 2 0 0 2 m 0 4 2 0 0 3 m 0 4 2 0 0 4 m 0 4 2 0 0 5 m 0 4 2 0 0 6 m 0 4 t std
Figure 3.3.2: Estimated standard deviations
For March 2007 the estimated variance equals 209.855 . Therefore, the values for the kerosene-gas-price-ratio for March 2007, computed for the simulation carried out in chapter five, are drawn from the following distribution:
kt=132
~ N (163.442, 209.855) (3.3.5)
nbpt=132
The notation t=132 refers to the month March 2007. This distribution is also used for the development of option strategies in chapter four.
3.4 An external factor model
As an alternative for the historical price equation a model is developed where external factors explain the one-month-ahead price ratio. A clarification of the selected external factors is provided in advance of the model description. External factors that are supposed to affect the kerosene price, gas price and both prices are discussed sequentially.
As kerosene is by far the most widely used jet fuel, passenger numbers and aviation accidents are collected as explanatory variables. Unfortunately only information on US instead of European passenger numbers could be collected. The assumption that US passenger numbers provide a good proxy for its European variant allows including this variable in the model. The data on aviation accidents consists of monthly-based data on the number of worldwide accidents with ten or more casualties.
The UK gas price is supposed to be related to supply and demand indicators. Therefore, data on the volumes of the UK production, import, export and storage of gas are included. Demand-related factors that are observed in the UK are temperature, industrial output, the gas consumption of power plants and the power price. The last two factors are taken in since gas is a major fuel for power plants in the UK. The information on temperature is gathered in Bedford, which is the most central weather station in the UK. The reason to take industrial output into the model is that the UK industry is, beside power plants, a major consumer of natural gas. Data on inflation is also collected since inflation is a well-known price-determining variable in over-the-counter gas contracts in the UK.
In addition, the prices of alternative fuels may affect both the kerosene and the gas price. Therefore, the prices of Brent crude oil, gasoil, fueloil and coal are included together with the price of gas traded in the US, known as Henry Hub. Some fuel consuming industries have to deal with regulations on the emission of carbon dioxide. Therefore, the price of CO2 emission rights is included. Since both kerosene and gas are commodities that may be considered as investment opportunities, it is reasonable to take information on alternative investments into the model. For this reason data on interest rates and indices of the Dow Jones and the FTSE 100 are included, where the FTSE index is converted from British Pounds into US Dollars. The exchange rate between US Dollars and British Pounds itself is also taken in. Further details on the collected data can be found in appendix C.
Similar to the historical price model in the previous section, this external factor model uses the kerosene-gas-price ratio as endogenous variable. This allows a good comparison between the two models. By explaining the kerosene-gas-price ratio of month t+1 with variables of month t predeterminedness of the explanatory variables is established which prevents endogeneity of the explanatory variables.
Since the gas price is the denominator of the fractional endogenous variable, it is expected that a model that includes the alternative fuel prices divided by the gas price yields a better fit than a model that includes the absolute prices. The same expectation holds for the price of CO2 emission rights, since the costs of CO2 emission rights can be interpreted as additional cost to the use of fossil fuels. An alternative model that contains the absolute prices of fuels and CO2 emission rights was also estimated but had a worse fit and is, therefore, not included in this thesis. The described model, which contains relative price of fuels and CO2 emission rights, is formulated in the following way:
kt+1
= c0 + c1 Passengers t + c2 Accidents t + c3 Gas production t + c4 Gas import t
nbpt+1
+ c5 Gas export t + c6 Gas storage t + c7 Temperature t
+ c8 Industrial output t + c9 PPGC t + c10 Power Index t + c11 Inflation t
+ c12 Dow Jones t + c13 FTSE t + c14 Exchange rate t + c15 LIBOR t
Brent t Gasoil t Fueloil t Coal t
+ c16 + c17 + c18 + c19
nbp t nbp t nbp t nbp t
Henry Hub t CO2 t
+ c20 + c21 + λ t+1 ~ NID(0, Ω 2
) (3.4.1) nbp t nbp t
with
c0 as a constant and c1 – c21 as coefficients;
λ t as the error term in month t;
Ω2
as the variance of the error t and
From section 3.2 the conclusion can be drawn that the kerosene-gas-price ratio is stationary. Although some of the explanatory variables, for example passenger numbers or industrial output, can be expected to be non-stationary, the formulation of (3.4.1) can be allowed by the realizing that cointegration among the explanatory variables can occur. As in model (3.3.1) the hypothesis that the standard errors of the estimators are distributed normally was not rejected, which justifies the use of maximum likelihood to estimate the parameters. It was tested whether the error terms are autocorrelated by using the Ljung-Box Q-statistic with twelve lags included. The hypothesis that the error terms of (3.4.1) do not show autocorrelation could not be rejected at a 5% significance level. With respect to heteroskedasticity of (3.4.1) an ARCH-LM test was performed up to twelve lags. The hypothesis that there is no autoregressive conditional heteroskedasticity in the residuals was not rejected at a 5% significance level. Therefore, the variance of the error terms is assumed to be independent of time. The conclusion can be drawn that the information provided by the explanatory variables eliminates volatility clustering as found in (3.3.4). Appendix D contains the numerical outcomes of the above mentioned tests. Estimating model (3.4.1) resulted in a substantial number of coefficients that were not significant at a 5% significance level using the t-statistic. Therefore, a procedure, based on the findings of David Hendry (2005) and Doornik (2007) was executed to exclude insignificant variables from the model.
The first step (A) of this procedure consists of excluding the most insignificant variable. This variable is chosen by selecting the one with the lowest absolute t-statistic and added to an artificial list of insignificant variables. At the next step (B) it is tested whether it is allowed to remove the variable selected at (A) from the model. It is tested whether the error terms of the remaining model, referred to as model (3.4.1 In progress), are not autocorrelated by using the twelve lags Ljung-Box Q-statistic. In addition it is tested whether the error terms of (3.4.1 In progress) are homoscedastic by using the ARCH-LM test up to the twelfth lag. Finally the hypothesis is tested whether it is allowed to set all coefficients of the artificial list of insignificant variables equal to zero is. This is tested by performing an F-test on the complete model (3.4.1). All hypotheses are tested at a 5% significance level. The variable selected at (A) is not excluded from the model if the hypothesis is rejected that the residuals are not autocorrelated, if homoscedasticity is rejected or if the hypothesis is rejected that it is allowed to set all coefficients of the artificial list of insignificant variables equal to zero. In all other cases the selected variable is excluded form the model. Subsequently the whole procedure is repeated considering the model
(3.4.1 In progress) until no insignificant variables are observed or the remaining insignificant variables cannot be excluded because of the outcome of tests at (B).
Executing this procedure for model (3.4.1) resulted in excluding the variables Accidents, Gas Import, Gas Storage, Power Index, Dow Jones, FTSE, Exchange rate and LIBOR together with the gas-price-ratio variables Gasoil, Fueloil, Coal and C02. The procedure was ended as there were no more insignificant variables. The remaining model is formulated in the following way:
kt+1
= 264.863 + 6.571·10-6 Passengers t + 1.743·10-7 Gas production t
nbpt+1 (77.861) (2.393·10 -6
) (6.227·10-8) - 4.639·10-7 Gas export t + 16.646 Temperature t
(1.488·10-7) (5.439)
- 1.135 Industrial output t - 8.223·10-7 PPGC t - 0.788 Inflation t
(0.441) (3.485·10-7) (0.291) Brent t Henry Hub t
+ 0.369 - 1.716 + λ t ~ NID(0, 136.532) (3.4.2)
(0.117) nbp t (0.553) nbp t
The series Brent/nbp and Henry Hub/nbp are stationary because of substitution effects between fuels. Temperature also shows stationary properties which can be explained by the repeating nature of a climate. However, all of the other variables in (3.4.2) are in general increasing in time, with some occasional exceptions, for example the temporary decrease in passenger numbers after 9/11/2001. The combination of positive and negative coefficients of these variables does not exclude cointegration which allows formulation (3.4.2).
Numerical information of the computed F, Ljung-Box Q and ARCH-LM statistic for this model can be found in appendix D, which also provides information on t-statistics of the estimated coefficients. The R-squared of this regression equals 0.895. For interpreting the coefficients the combined effect of variables on both kerosene prices and gas prices should be considered simultaneously. Passenger numbers have a positive effect on the next-month price ratio. An explanation can be that a higher passenger number increases the month-ahead demand and consequently the price of kerosene, without affecting the gas price. The price of kerosene can be assumed to be independent of the gas production, gas export, temperature, industrial output, gas consumption by power plants and inflation in the UK. Gas production increases the supply of gas
and can, therefore, be expected to affect the month-ahead gas price negatively and the ratio positively. The opposite holds for gas export. An increasing temperature lowers the demand for gas and consequently the price, which explains the positive effect on the price ratio. Industrial output and gas consumption by power plants increase the demand and the price of gas, which let the price ratio decline. Inflation is taken in most over the counter gas contracts as a price-increasing factor. This clarifies the negative coefficient of this variable. With respect to the gas-price-ratio variables Brent and Henry Hub the following explanation can be given. Although Brent is a price driver for both gas and kerosene its link with kerosene is stronger since kerosene is an oil distillate and gas is only an alternative fuel. This explains the positive coefficient in the model. The US Henry Hub gas is a price indicator for gas in the UK. An increasing relative price of Henry Hub will in general affect UK month-ahead gas prices positively, which let the ratio between kerosene and gas prices decline.
Figure 3.4.1 plots the actual values of the kerosene-gas-price ratio and the fitted values of the external factor model (3.4.2).
Figure 3.4.1: Actual and fitted values of model 2 for the kerosene-gas-price ratio
As equation (3.4.2) shows there are differences in the order of magnitude of the estimated coefficients. This is caused by differences in the order of magnitude of the variables. For example the average monthly total number of passengers is almost eleven million compared to a round off
Ratio kerosene and gas prices
0 50 100 150 200 250 1 9 9 6 m 0 4 1 9 9 7 m 0 4 1 9 9 8 m 0 4 1 9 9 9 m 0 4 2 0 0 0 m 0 4 2 0 0 1 m 0 4 2 0 0 2 m 0 4 2 0 0 3 m 0 4 2 0 0 4 m 0 4 2 0 0 5 m 0 4 2 0 0 6 m 0 4 t kerosene/nbp actual values model fitted values
average temperature of nine degrees. The absolute value of the average addition of each variable to the price ratio of (3.4.2), computed by multiplying the average value of the variable with the absolute value of its coefficient, lies between 13 and 156. As mentioned in (3.3.3) the round off average value of the kerosene-gas-price ratio is 126.
It should be noted that another model was estimated where all gas related factors of (3.4.1) were converted to reciprocal values. This conversion was executed since the endogenous variable, the kerosene gas price ratio, also contains the reciprocal value of the gas price. The above-described procedure was executed for this model. However, since its fit was worse than the fit of model (3.4.2), the model with reciprocal values is not taken in this thesis.
By using (3.4.2) together with the variable values of February 2007, the last month of the dataset, an expected value of the kerosene-gas-price ratio for March 2007 is computed. The value of this forecast equals 176.423 . Therefore, the values for the kerosene-gas-price-ratio for March 2007, computed for the determination of option strategies in chapter four and simulations carried out in chapter five, are drawn from the following distribution:
kt=132
~ N (176.423, 136.532) (3.4.3)
nbpt=132
Remark that the variance of this distribution was found at the estimation of model (3.4.2).
3.5 A kerosene price model
Since models that explain the price ratio do not provide information on the absolute value of the kerosene price and the gas price, a model is needed that forecasts one of these prices. By using the forecasted ratio it is possible to obtain information on the price of the other fuel. Similar to the model described in section 3.3 the model developed in this section explains the one month-a-head price by using historical realisations of the price. However, since a single price instead of a ratio between the prices of two goods is observed, a model that explains the log of the ratio between two successive monthly average prices is more appropriate. A model that explains the ratio
instead of the log ratio was also estimated. Since the explanatory power of the log linear model in terms of R2 is higher this model is selected. A pure geometric Brownian motion could be used; however, this is most useful for generating long-term price paths. Since the purpose of the model in this thesis is to give a one-month-ahead price forecast a model is preferred that pays more attention to the short-term history. Therefore, an ARIMA-GARCH specification for log kt is used.
Estimating the price models for both fuels showed that the kerosene price model has a better fit which makes that this model is used in the thesis. The following specification holds for this model: kt m kt-i p log = µ + Σ ai (log – µ) + Σ bjεt-j + εt , εt ~ N(0, σ 2 t) (3.5.1) kt-1 i=1 kt-1-i j=1 q r σ2t = ω + Σ ciεt-i2 + Σ djσ2t-j (3.5.2) i=1 j=1 with
kt as the price of kerosene in month t;
m as the number of included autoregressive terms; p as the number of included moving average terms; ai as the coefficient of the i-th autoregressive term;
bj as the coefficient of the j-th moving average term;
εt as the error term in month t;
σ2
t as the estimated variance of the distribution of the error term ε t;
ω as a constant;
q as the number of included squared error terms; r as the number of included estimated variance terms; ci as the coefficient of the i-th squared error term and
dj as the coefficient of the j-th estimated variance term.
As estimator for µ the sample mean of the logs of the ratios of kerosene prices of two succeeding months is taken. Maximum likelihood was used to estimate the parameters as the hypothesis that the standard errors of the estimators are distributed normally was not rejected. Numerical outcomes of the Jarque-bera test for normality and other tests announced later on in this section are provided in appendix E. The parameter values of m, p, q and r are determined exactly the
same as described in section 3.3. Estimation leads to an ARMA(1,1)-GARCH (1,1) model described below. kt kt-i log = µ -0.473 (log - µ ) + 0.736 εt-1 + εt , εt ~ N(0, σ 2 t) (3.5.3) kt-1 (0.153) kt-1-i (0.233) µ = 0.009076 (3.5.4) σ 2t = 7.541·10-5 + 0.0623 ε t-1 2 + 0.909 σ 2t-1 (3.5.5) (3.596·10-5) (0.024) (0.316)
Appendix E gives information on t-statistics of the estimated coefficients. The expectation of the log kerosene price ratio March 2007/February 2007 can be computed by using (3.5.3) and (3.5.4), which leads to a value of -0.024742 . For the estimated GARCH (1,1) model it holds that c1 + d1
= 0.062310 + 0.9094 = 0.9717 < 1, so the unconditional variance exists and equals ω /(1- c1 -
d1) = 7.541·10-5 / (1 - 0.0623 - 0.909) = 0.002666. For March 2007 the estimated variance equals
0.003187. In addition the average price of a barrel of kerosene in February 2007 was USD 74.54. Therefore, in chapter four and five the following distribution for the price of kerosene in March 2007 is used: kt=132 = 74.54 e z , z ~ N (- 0.024742, 0.003187) = ey , y ~ N (- 0.024742 + log 74.54, 0.003187) (3.5.6)
To conclude this chapter it should be mentioned that the correlations between the error terms of the models (3.3.3), (3.4.2) and (3.5.3) are between -0.02 and 0.01. Because of these low values, they are assumed to be zero in the rest of the thesis.
4 Option strategies
The price models of chapter three provide insight in the distributions of the future kerosene and gas prices. The realised prices affect the financial performances of agents who trade in kerosene or gas. The portfolios of agent Alpha and Beta are considered, where Alpha is a consumer of kerosene and Beta produces natural gas. Alpha is exposed to price risks with respect to his expenditures on kerosene. Beta on the other hand has to deal with price uncertainty concerning the revenues of his sales of gas. In this chapter several option strategies are developed that decrease the risks of both agents. At first in section 4.1 mutual option contracts are designed between Alpha and Beta. Then in section 4.2 the most efficient option contracts of the mutual contracts are transformed to third-party contracts. Finally a parameter hedge is implemented in section 4.3.
4.1 Mutual option contracts
As can be deduced form the introduction of this chapter, Alpha will see its cost on kerosene rise if the price of kerosene increases. By buying call-options on kerosene this risk can be reduced. Buying call-options requires capital. Furthermore, some agents have the obligation to report the value of their assets to their shareholders. Although call-options can reduce price-risks, they cause volatility on the asset value reports until the moment they are exercised. By selling put-options on kerosene the required capital for hedging and the volatility of the asset value report are both reduced. Hedging strategies that consist of buying a call-option (put-option) and selling a put-option (call-option) are also referred to as collar spreads. Figure 4.1.1 provides an example of a collar spread for a kerosene-consuming agent.
Collar spread: buying kerosene and a call-option, selling a put-option
0 20 40 60 80 100 120 0 10 20 30 40 50 60 70 80 90 100
Kerosene price (USD) Value (USD)
Kerosene (1 barrel) Call-option (strike=70) Put-option (strike=50) Expenditures for total portfolio
Figure 4.1.1: A collar spread option strategy for a kerosene consumer
A similar figure can be drawn for a gas-producing agent. It should be remarked that the kerosene is bought at the moment that the options expire. Furthermore the expenditure on the purchase of the options and income from the sale of the options are not included; the lines in the figure represent the pay-off of the options at the realised kerosene and gas prices. A special case occurs if the strike of the put and the call-option are equal; a so-called future on kerosene is then created. Collar spreads are more attractive than futures for hedging purposes since futures are relatively expensive compared to collar spread structures. Therefore, collar spreads instead of futures will be considered in this thesis.
As Alpha consumes kerosene and Beta produces gas, the implemented mutual option strategy consists of hedging Alpha’s costs on kerosene with both options on kerosene and gas. From the perspective of risk management, the size of the part of the cost on kerosene that is hedged with kerosene options and the size of the remaining part that is hedged with options on gas should be determined by observing the risk premiums on options on both fuels. As there is no listed option market for both kerosene and nbp gas and over the counter contracts are strictly confidential, there is no information on risk premiums. For the determination of the part of Alpha’s cost on kerosene that is hedged with options on kerosene and the part that is hedged with gas options, it is assumed that the premiums on options on kerosene and option on gas are equal. This leads to hedging half of Alpha’s costs on kerosene with options on kerosene; the other half is hedged with options that have gas as underlying. In addition half of Beta’s gas sales are hedged with options on gas while for the remaining half options on kerosene are made use of. The expectations of the kerosene-gas-price ratio deduced in chapter three at (3.3.5) and (3.4.3) provide a forecast of the number of terms of gas that is expected to have the same value as one barrel of kerosene in March 2007. By the lack of information on risk premiums, the expectation of the kerosene-gas-price ratio also represents the number of options on gas that is expected to provide protection to the cost of one barrel of kerosene in March 2007.
If Alpha sells Beta the put-options requested by Beta and Beta sells Alpha the call-options requested by Alpha, mutual transactions are created that lead to collar spread option strategies for both agents. This transaction is visualised in schedule 4.1.2:
sale of put-options on kerosene and gas
sale of call-options on kerosene and gas
Schedule 4.1.2: Mutual transactions between Alpha and Beta creating collar spreads for both agents
Since two different models are estimated with respect to the kerosene-gas-price ratio, two option strategies are developed with different strikes. The following table summarises the different strikes that correspond to the options of the mutual transactions in schedule 4.1.2:
Model Option
historical price model external factor model
put on kerosene S(P, k, H) S(P, k, F)
put on gas S(P, nbp, H) S(P, nbp, F)
call on kerosene S(C, k, H) S(C, k, F)
call on gas S(C, nbp, H) S(C, nbp, F)
Table 4.1.3: Abbreviations of considered options
The abbreviation S(x, y, z) represents the strike (S) of an x-type option on y, using model z. The type of option can be put (P) or call (C); the underlying can be kerosene (k) or gas (nbp) and the chosen model is the historical price model (H) or the external factor model (F).
As it is the purpose of the option strategy to reduce the price risk in March 2007, all options of the mutual transactions expire in March 2007. The strikes of the options are chosen such that fluctuations with a negative impact are cut-off at a one percent level from its expected value. In reality higher percentages may be more appropriate. However, since price models are estimated in this thesis relatively small confidence intervals of the future prices are obtained. Option strategies that protect against for example a 5% price change will seldom result in a pay-off of the options when prices are simulated with the models. Therefore, option strategies that protect against big price changes only are less suitable to evaluate by simulation than strategies that protect against small price changes. On average, the probability that an option of a strategy that protects against 1% price changes and higher leads to a pay-off is 0.41. This relatively high probability increases the solidity of a comparison between option strategies based on simulations.
The option strategy that protects against 1% price changes implies that the strike of the call-options on kerosene bought by Alpha from Beta have a strike that is one percent higher than the expected kerosene price for March 2007; the strike of the put-options on gas sold by Alpha to Beta have a strike that is one percent lower than the expected gas price for March 2007. In terms of the notation used in chapter three this leads to the following equations:
∞ E[kt=132] = ∫ k f(k) d k = 74.54 · exp(µ +½ σ 2 ) = 72.834 (4.1.1) 0 A ∞ ∞ ∞
EH[nbpt=132] = ∫ ∫ (k/ratio) f(k,ratio) d k d ratio + ∫ ∫ (k/ratio) f(k,ratio) d k d ratio
-∞ 0 B 0
= 0.44922 (4.1.2)
A ∞ ∞ ∞
EF[nbpt=132] = ∫ ∫ (k/ratio) f(k,ratio) d k d ratio + ∫ ∫ (k/ratio) f(k,ratio) d k d ratio
-∞ 0 B 0 = 0.41469 (4.1.3) S(C, k, H) = S(C, k, F) = 1.01 E[kt=132] = 73.563 = S(C, k) (4.1.4) S(P, nbp, H) = 0.99 EH [nbpt=132] = 0.44473 (4.1.5) S(P, nbp, F) = 0.99 EF [nbpt=132] = 0.41054 (4.1.6) with
E[kt=132] as the expected kerosene price for March 2007;
EH[nbpt=132] as the expected gas price for March 2007 according to the historical (H)
price model;
EF[nbpt=132] as the expected gas price for March 2007 according to the external factor
(F) price model;
f(k) as the distribution function for the price of kerosene derived from (3.5.6); f(k,ratio) as the joint distribution function for the price of kerosene and the price
ratio, derived from (3.3.5) and (3.5.6) for EH[nbpt=132] and derived from
(3.4.3) and (3.5.6) for EF[nbpt=132];
A as the limit ratio ↑ 0 and B as the limit ratio ↓ 0.
The fraction k/ratio in (4.1.2) and (4.1.3) represents the nbp gas price as can be deduced from (3.3.5), (3.4.3) and (3.5.6). The integrals of (4.1.2) and (4.1.3) are split up with limits A and B in
order to prevent that the denominator of the fraction k/ratio can take the value zero. The normal distribution of the kerosene-gas-price ratio attaches positive probabilities to negative ratio values. This leads to a very small probability of negative gas prices, which is supported by historical realised gas prices. The first equality in (4.1.4) holds since the kerosene price is not affected by the choice for the kerosene-gas-price ratio model. Therefore, the strike of a call-option on kerosene will be referred to as S(C, k).
With respect to the strike of the call-options on gas that Alpha buys (S(C, nbp, H) or S(C, nbp, F)) different choices can be made. The simplest method is to choose a strike that is one percent higher than the expected gas price for March 2007. A reason to choose for this approach is the fact that a kerosene-gas-price ratio model provides a forecast of the future relation between the prices between the prices of these fuels. From the perspective of the ratio model it is, therefore, expected that the price of gas will increase with the same percentage as the price of kerosene such that the ratio is unchanged. This method leads to the strikes
S(C, nbp, H) = 1.01 EH[nbpt=132] = 0.45371 and (4.1.7)
S(C, nbp, F) = 1.01 EF[nbpt=132] = 0.41883 . (4.1.8)
A different approach focuses on the expected pay-off of options. Alpha buys call-options on kerosene in order to get a compensation for the extra cost on kerosene in case the price exceeds the 101% level of the expected value of the kerosene price. The ideal call-option on gas for Alpha returns exactly the same pay-off as the call-option on kerosene does. Unfortunately, prices of kerosene and gas are not fully correlated. However, the strike of the call-options on gas can be chosen such that these gas options have the same expected pay-off as the call-options on kerosene have in case the price of kerosene exceeds the 101% level of the expected value. A call-option on gas that has as a strike developed in this way does not have the same pay-off as a call-option on kerosene for every combination of gas and kerosene prices. However, on average, for all price scenarios where the kerosene price exceeds the 101% level of the expected value, the pay-offs of the call-options on gas and kerosene are equal. The number of options on gas that is expected to be equivalent to one option on kerosene should be taken into account for this derivation. As mentioned before this number equals the expected value of the kerosene-gas-price ratio for March 2007. For the historical price model the derivation of the strike of a call option on gas can be expressed mathematically with the following equation:
∞
E[P(S(C, k))] / E[ratio] = ∫ max(0, k – S(C, k)) f(k) d k / E[ratio] =
0
∞ must
= ∫ (k – S(C, k)) f(k) d k / 163.442 = 1.3096 / 163.442 = 0.008013 =
S(C, k)
A ∞
= ∫ ∫ max (0, (k/ratio) - S(C, nbp, H)) I(k >S(C, k)) f(k,ratio) d k d ratio
-∞ 0 ∞ ∞
+ ∫ ∫ max (0, (k/ratio) - S(C, nbp, H)) I(k >S(C, k)) f(k,ratio) d k d ratio
B 0
A ∞
= ∫ ∫ max (0, (k/ratio) – S(C, nbp, H)) f(k,ratio) d k d ratio
-∞ S(C, k)
∞ ∞
+ ∫ ∫ max (0, (k/ratio) – S(C, nbp, H)) f(k,ratio) d k d ratio (4.1.9)
B S(C, k)
with
I(k>S(C, k)) as the indicator function that yields 1 if k > S(C, k) and 0 else and
E[P(S(x, y, z))] as the expected pay-off in March 2007 of the option with strike S(x, y, z).
Solving (4.1.9) leads to following value of S(C, nbp, H):
S(C, nbp, H) = 0.47061 (4.1.10)
If the external factor in stead of the historical price model is selected, equation (4.1.9) should be modified with respect to the expectation of the ratio (which is then 176.423 in stead of 163.442) and the joint distribution function f(k,ratio) should then be derived from (3.4.3) and (3.5.6) in stead of (3.3.5) and (3.5.6). This leads to following value of S(C, nbp, F):
S(C, nbp, F) = 0.42749 (4.1.11)
With respect to the strike of the put-option on kerosene that Beta buys (S(P, k, H) or S(P, k, F)) again different choices can be made. Choosing a strike that is one percent lower than the expected kerosene price for March 2007 leads to the following value:
The alternative approach, that focuses on the expected pay-off of options, finds a strike of the put-option on kerosene such that this put-option has the same expected pay-off as the put-put-options on gas in case the price gets less then the 99% level of the expected value. For the historical price model this relation can be expressed with the following equation:
A ∞
E[P(S(P, nbp, H))] · E[ratio] = [ ∫ ∫ max (0, S(P, nbp, H) - k/ratio) f(k,ratio) d k d ratio
-∞ 0
∞ ∞
+ ∫ ∫ max (0, S(P, nbp, H) - k/ratio) f(k,ratio) d k d ratio] · E[ratio] =
B 0
must
= 0.016683 · 163.442 = 2.7267 = A ∞
= ∫ ∫ max(0, S(P, k, H) - k) I(k/ratio < S(P, nbp, H)) f(k,ratio) d k d ratio
-∞ 0
∞ ∞
+ ∫ ∫ max(0, S(P, k ,H) - k) I(k/ratio < S(P, nbp, H)) f(k,ratio) d k d ratio =
B 0 A S(P,k,H)
= ∫ ∫ (S(P, k, H) - k) I(k/ratio < S(P, nbp, H)) f(k,ratio) d k d ratio +
-∞ 0
∞ S(P,k,H)
+ ∫ ∫ (S(P, k, H) - k) I(k/ratio < S(P, nbp, H)) f(k,ratio) d k d ratio (4.1.13)
B 0
Solving (4.1.13) leads to following value of S(P, k, H):
S(P, k, H) = 76.477 (4.1.14)
In case the external factor instead of the historical price model is selected, equation (4.1.13) should be modified with respect to the expectation of the ratio and the joint distribution function. These modifications are similar to the ones performed to obtain (4.11) from (4.9). This leads to following value of S(P, k, F):
S(P, k, F) = 75.049 (4.1.15)
In equations (4.1.1) – (4.1.15) all strikes of table 4.1.3 are collected. These options, which Alpha and Beta sell to each other, also have to be priced. As the distributions of the prices of kerosene and gas are known, pricing can be done by using integrals instead of standard option pricing
models. Agents are assumed to charge no transaction fees or margins to each other. If they would charge fees, they would act as a third party and a comparison between mutual and third-party contracts could not be made. Furthermore, if both agents would charge fees the fees would mostly cancel out at the net payment to each other. As mentioned before there is no information on risk premiums. To make a reasonable comparison between mutual and third-party options, risk premiums are, therefore, excluded from the price of both type of options.
Since the agents do not charge transaction fees, margins and risk premiums to each other, the price of an option equals its expected pay-off at expiration. In order to take into account the principle of discounting future pay-offs, the prices of the options are paid at the expiration in March 2007. This way of paying excludes the effect of interest rates.
The integrals that determine the prices of the considered options are formulated as follows: S(P, k, z) E[P(S(P, k, z))] = ∫ (S(P, k, z) – k) f(k) d k (4.1.16) 0 ∞ E[P(S(C, k, z))] = ∫ (k – S(C, k, z)) f(k) d k (4.1.17) S(C, k, z) A ∞
E[P(S(P, nbp, z))] = ∫ ∫ max (0, S(P, nbp, z) - k/ratio) f(k,ratio) d k d ratio
-∞ 0
∞ ∞
+ ∫ ∫ max (0, S(P, nbp, z) - k/ratio) f(k,ratio) d k d ratio (4.1.18)
B 0
A ∞
E[P(S(C, nbp, z))] = ∫ ∫ max (0, k/ratio - S(C, nbp, z)) f(k,ratio) d k d ratio
-∞ 0
∞ ∞
+ ∫ ∫ max (0, k/ratio - S(C, nbp, z)) f(k,ratio) d k d ratio (4.1.19)
B 0
with z as possible choice for the kerosene-gas-price ratio model; z can be H (historical price model) or F (external factor model). This choice affects the distribution function f(k,ratio).
The computed strikes and prices of all options of table 4.1.3 are collected in appendix F. A distinction is made between the methods to choose the strike of call-options on gas and put-options on kerosene. The put-options developed with the methods used in (4.1.7), (4.1.8) and (4.1.12)
are be referred to as levelled options. Options with strikes designed as in (4.1.9) – (4.1.11) and (4.1.13) – (4.1.15) are categorised as adjusted options. The strikes and prices of the options are expressed in five significant digits. This allows an accurate comparison of the pay-off of the options at the simulated price scenarios in chapter five.
With the information of the tables F.1 and F.2 in appendix F a complete overview of all mutual transactions can be obtained. As explained earlier, the expected value of the kerosene-gas-price ratio is a determinant factor for the transactions as well. It is assumed that Beta buys 163.442 put-options on gas from Alpha, sells 2·163.442 terms of gas at the market and sells 163.442 call-options on gas to Alpha in March 2007 when the historical price model is selected. Furthermore, Alpha buys one call-option on kerosene from Beta, buys two barrels of kerosene at the market and sells one put-option on kerosene to Beta. The number of terms of gas is multiplied by two since half of the sale of gas is hedged with 163.442 put-options on gas; the other half is hedged with one put-option on kerosene. The same holds for the amount of barrels of kerosene, which therefore equals two.
In case the external factor model is selected Beta is assumed to buy 176.423 put-options on gas from Alpha, to sell 2·176.423 terms of gas at the market and to sell 176.423 call-options on gas to Alpha in March 2007 if the external factor model is selected. Alpha again buys one call-option on kerosene from Beta, buys two barrels of kerosene at the market and sells one put-option on kerosene to Beta. This leads to the following transactions in March 2007:
max(0, S(P, k, z) - kt=132 )
E[Ratio]·max(0, S(P, nbp, z) - nbpt=132)
net payment for purchase of options
max(0, kt=132 - S(C, k, z))
E[Ratio]·max(0, nbpt=132 – S(C, nbp, z))
2·kt=132 2·E[Ratio] terms of gas
2 barrels of kerosene 2·E[Ratio]·nbpt=132
Schedule 4.1.6: Transactions between Alpha, Beta and the market
With
z as the selected kerosene-gas-price ratio model;
kt=132 as the realised kerosene price in March 2007;
nbpt=132 as the realised nbp price in March 2007, nbpt=132 = kt=132/ratiot=132 and
E[Ratio] as the expected value of the kerosene-gas-price ratio.
The net payments for the purchase of options consists of the price of one call-option on kerosene, plus the expected value of the price ratio times the price of one call-option on gas, minus the price of one put-option on kerosene, minus the expected value of the price ratio times the price of one put-option on gas. Schedule 4.1.6 is drawn in appendix F.3 – F.6 for the four combinations of the kerosene-gas-price ratio model (historical or external factor) and the type of options (levelled or adjusted). This completes the design of the mutual option strategies when the models that describe the kerosene-gas-price ratio are not modified with respect to the correlation between the prices of both fuels.
However, as described in the research question in the introduction of this thesis, it is also researched how the rational choice between mutual option contracts on price-correlated products and third-party contracts on a single product depends on price-correlation. In order to get insight in this relation, the same option strategies computed in (4.1.1)-(4.1.15) are developed in case the
Alpha Beta
