Investment in Gas Field Depletion

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Investment in Gas Field Depletion

A Real Options Approach

Master Thesis

Author: Xinyu Zhang Student Number: S3489914

Supervisor: Dr. G.T.J. Zwart

Msc. Finance

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Abstract

The main research question in this paper is to determine the optimal timing of the investment in gas field depletion. The real option method is applied to model the value the investment of this project. The only uncertainty as a stochastic variable of this model is the price of natural gas which follows the geometric Brownian motion. The process of this investment is divided into two stages, so the goal of the model is to find trigger prices for each stage. To solve this problem, codes for the option by explicit finite difference method are ran on Python. A numerical example is taken based on data of the Dutch gas market. The result shows that the threshold at the second stage is lower than that at the first stage, meaning that the investor will waiting longer for the start of the investment.

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1. Introduction

The gas market in Europe is facing various challenges because of the decreasing gas reserve and changes of gas market conditions. The demand of natural gas is expected to rise, while the production is decreasing.As the main supplier in the Europe, the Netherlands has seen a 39.8% decline of proved natural gas reserves from 2004 to 2014 due to long period significantly gas production. The Netherlands also face the stress of importing more natural gas. The risks arises when relying on foreign imports due to the undulations of gas market and inflexible supply.

As a kind of non-renewable resource, natural gas has the finite reserve. Therefore the gas field depletion path should be carefully planned. The main research problem of this thesis is to find the optimal timing of investment on the gas field depletion. The trigger price should be estimated to decide the investment timing. To calculate the threshold, the real option method is used, since it covers the main characteristics of the investment in gas field depletion: the nonreversibility of the investment, payoff uncertainties, and the flexibility of investment timing (Dixit and Pindyck, 1994).

This paper follows the real option framework from Dixit and Pindyck (1994). Consider a company has the right to invest in the gas field depletion immediately or postpone the investment. The only uncertainty as a stochastic variable in this model is the gas price, which follows the geometric Brownian motion. The investment process is divided into two stages, of which the trigger prices should be estimated.

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2. Literature overview

2.1 Research on depletion

Some attention has been paid on the depletion of finite resource, especially on the oil and gas. Wakeford (2012) analyses the socioeconomic implications of global oil depletion for South America. Based on the historical evidence and empirical models, he concludes that future oil shortage will induce a gradual constriction in the economy by increasing unemployment and inflation, or in the worse case, lead to a systematic collapse of crucial infrastructure systems. The suggestions to mitigate are provided, such as investing in renewable energy and reforming monetary system. Posner (1972) investigates when to deplete or use up the gas reserves under the North Sea in the late 1960s. He conclude that under certain assumptions1_{, the current round of natural gas}

contracts should involve the rapidest possible depletion.

There are some articles studying the production and demand of natural gas, as well as their factors. Attanasi (1986) focuses on the industry operating in the offshore Gulf of Mexico. The conclusion of this paper is that the future production of oil and gas, and the offshore drilling and production facilities demand, are dependent on the undiscovered field size distribution, joint production costs, and prices of oil and gas. Moreover, with the developing and using the cost reducing technologies, there could be a massive payoff to the drilling industry, in term of the increasing potential demand. It also discusses implications of these conclusions for other offshore producing areas like the North Sea.

2.2 Research applying real options

The study on the evaluation with uncertainties and the importance of investment timing began in the 1980s (Brennan and Schwartz, 1985; McDonald and Siegel, 1986). Brennan and Schwartz (1985) realise that it is difficult to value the mining and other

1_{Posner gives three points that suggest more rapid depletion: Firstly, assume that no more gas would be found in}

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natural resource projects due to the uncertainty of output prices. This paper shows the way to evaluate a project whose cash flows is dependent on highly mobile output prices and determine the optimal policies for managing them is to exploit the properties of replicating self-financing portfolios. One of the earliest research on the optimization of investment timing was written by McDonald and Siegel in 1986. They originate the rule of optimized investment and build and model to estimate the value of the investment option. The conclusion of their study is that the right for a investor to choose when to invest leads to a higher value of the project.

Dixit and Pindyck (1994) provide and explain a framework of real options, which is applied in a number of research articles in energy field as well as other scientific areas. The real option method takes several considerations on nonreversibility of the investment, payoff uncertainties, and the flexibility of investment timing (Dixit and Pindyck, 1994).

The real option theory is applied in several articles (Murto and Nese, 2003; Siddiqui and Maribu, 2009; Fan and Zhu, 2010). Murto and Nese (2003) apply the real option framework from Dixit and Pindyck (1994) to build a dynamic model of an investment decision between fossil fuel and biomass fired production technologies. They find that if the choice of technology is nonreversible, it is optimal to wait instead of invest immediately even if that it is optimal to invest in one or both type of plants. Siddiqui and Maribu (2009) analyze on the a microgrid that can reduce its risk on gas price volatility by proceeding with distributed generation capacity and heat exchangers investment in a sequential manner of investment. The conclusion of this paper is that for low levels of gas price volatility, the microgrid prefers to a direct investment strategy, and for higher levels of volatility, it prefers to a sequential one. The real option model is also applied by Fan and Zhu (2010) and developed to elucidate how to evaluate and compare the oil-resource investment value in diverse countries under the price of oil, exchange rate, and uncertainties of investment environment. They identify the option value index to compare oil-investment conditions in different countries and prove that this extended model can provide some helpful suggestions for China’s overseas oil investment program.

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al., 2006; Wang, et al., 2013). The real options method is an advantageous tool for optimal decisions on investment and production behaviour, because it combines the market uncertainty, business environment, and flexibility of management.

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3. Introduction to natural gas

In this section, we will get the overview of natural gas and its market, including the properties of natural gas, demand and supply of natural gas, and the gas market in the Netherlands.

3.1 Properties of natural gas

According to The Economic Times, natural gas is a gaseous fossil fuel consisting mainly of methane, commonly including other kinds of higher alkanes, also a tiny proportion of carbon dioxide, nitrogen, hydrogen sulfide, or in some cases, helium. Natural gas is mainly found in oil fields and natural gas fields, and there are also a few out of coal seams. As a fossil fuel, natural gas can be used for power generation and heating, or as a fuel for natural gas vehicles. Among all hydrocarbon energy sources, natural gas is the cleanest and the most hydrogen-rich product (Economides and Wood, 2009). Compared to other kind of fossil energy, combusting natural gas produces less coom and carbon dioxide (Heeswijk, 2012).

3.2 Demand of natural gas

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until 1994, then, oscillated between 600 bcm and 700 bcm. In the rest of the world, the average growth rate of the demand is 6.1% per year. Especially in China, the growth rate is 12.8%, which is even higher.

Figure 3.1. World natural gas demand by selected regions

Gas market report 2017 shows that the growth rate of the demand will be 1.6% per year in the five-year forest to 2022 by IEA. In this case, the consumption of the natural gas will grow from 3.63 thousand bcm in 2016 to four thousand bcm in 2022. Nearly 90% of the expected growth comes from developing countries, led by China. IEA mentioned that half of the anticipated natural gas consumption is from the industry. Specifically, the annual growth rate of industrial gas demand is about 3% because of the increasing consumption of natural gas in chemical industry, fertilizer demand in countries such as Indonesia and India, and the substitution of natural gas for coal in small industrial sectors in China. The gas consumption for transportation will reach 140 bcm in 2022, while it was 120 bcm in 2016. As the main part of gas demand, power generation carries on with a positive but gentler growth rate (less than 1%). However, there is increasing power generation based on renewable energy and a lower growth rate of the electricity demand, which bounds opportunities for thermal power plants in mature markets. Natural gas also faces a strong competition from coal in emerging markets that rely on imported gas and have less limitation on air pollution.

3.3 Supply of natural gas

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exploring and producing technology. According to BP (2017), the total world proved reserves in 2016 was 186.6 trillion cubic metres (tcm), while that in 2006 was 158.2 tcm. The majority of the proved reserves were in the Europe & Eurasia (56.7 tcm) and Middle East (79.4 tcm).

World Energy Council (2016) shows that the average growth rate of natural gas production was 3.3% between 2009 and 2013, and slowed down because of the drop of oil and gas prices in 2014. The production was 3538.6 bcm in 2015, with 2.2% growth from that in 2014. The production in 2016 was 3551.6 bcm (BP 2017).

According to World Energy Resources: Natural Gas 2016, the main natural gas producer in the world are the United States (767.3 bcm), Russia (573.3 bcm), and Iran (189.4 bcm). As a result of shale gas exploitation in the US, North America witnessed a remarkable growth of 3.9%, representing 28.1% of the worldwide natural gas production in 2015. The natural gas production in Latin America and Caribbean grew only 0.7% and reached 178.5 bcm in 2015, while the average growth rate was 1.6% per year during 2010-2014. A lack of investment in upstream operations resulted in the drop of natural gas production in Europe, which declined with 5% annually at average in the last ten years. The production in Europe was 132.4 bcm in 2014 with a 9% decrease from 2013. Russia produced 573.3 bcm natural gas in 2015 with a decline of 1.5% over 2014. The negative growth rate of natural gas production since 2013 was due to the sanction, price reduction, and global demand decrease. The production in the Middle East was and will be growing recently. The average annual growth rate was 4.9% between 2010 (495.6 bcm) and 2014 (599.1 bcm). WEC shows that as the largest producer in the Middle East, Qatar, Iran, and Saudi Arabia are expected to continue to increase production until 2020. Africa has seen a reduction of natural gas production in 2013-2014, and a slight grow in 2015, reaching 211.8 bcm from 208 bcm in 2014. The discovery of large offshore around East Africa (especially Mozambique and Tanzania) leads to a positive expected growth rate of natural gas production in Sub-Saharan Africa. The production in Asia Pacific was 556.7 bcm. This represented a growth of 4.1% year-over-year, which was higher than the average growth rate (1.8%) between 2010 and 2014. Chinese production increased in 6.8% average annual rate during 2009-2015 and reached 138 bcm in 2015, which represented a large proportion of the growth in Asia Pacific.

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gas demand are not the same, therefore the transportation and storage of natural gas is necessary (Rumbauskaitė, 2011).

3.4 The gas market in the Netherlands

After the first discovery of natural gas in 1948 (Correlje, Van der Linde, and Westerwoudt, 2003), the Netherlands started to play a key role in the gas market. Honoré (2017) shows that the Netherlands has the fifth largest gas market in Europe (next to Germany, the United Kingdom, Italy, and Turkey). It is the second largest regional gas supplier and exporter behind Norway. To boost production from small field and attract investment of it, the Netherlands has been using the “small field’s policy” and leave the large field in Groningen to provide for “swing supply” (Rumbauskaitė, 2011).

Data from Central Bureau of Statistics (CBS) shows that the consumption of natural gas in the Netherlands reached 41.1 bcm in 2017, decreasing from 53.0 bcm in 2010 when the demand in the whole Europe was singularly high because of both economic recovery after 2009 and the unconventionally cold weather (Honoré, 2017). According to CBS, the total final consumption of natural gas in 2016 was 735.1 Petajoules (PJ), including final energy consumption (653.4 PJ) and non-energy consumption (81.7 PJ). The main demand of natural gas came from dwellings sector (297.2 PJ) and industry (excluding the energy sector) which was 172.3 PJ. Honoré (2017) argues gas demand on dwellings is temperature sensitive, because 98% of households consume gas to cook and heat. The main part of industrial gas consumption was from the chemical and petrochemical sector (61.8 PJ).

The Indigenous natural gas production in Netherlands was about 43.91 bcm in 2016, accounting for 107.1% of the national demand (CBS). According to the World Energy Council (2016), long time large production of natural gas has led to the decline of proved natural gas reserves in the last few years. There was a specifically significant drop off in natural gas production in 2014 when the production in the Netherlands was 55.8 bcm, representing an 18.7% decrease compared to previous years. In spite of the large decline in production, the Netherlands was still the leading natural gas producer in the EU and the 15th_{largest worldwide producer in 2014. With the limited quantity of}

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4. Real option methodology

The previous section provided an overview of global and Dutch gas market. In this section, we turn to introduce the real option theory for analytical part of the paper, including the overview and drawbacks of traditional valuation methods, the real option analysis (the superiority, mathematical background and valuation method), and the trinomial tree model.

4.1 Traditional valuation methods

There are many methods to measure the value of the investment. Drury (2008) distinguishes four main traditional valuation methods: the Accounting Rate of Return (ARR), the payback method, the discounted cash flow analysis (DCF), and the Internal Rate of Return (IRR).

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IRR is that it lacks flexibility to new information. Another drawback is that it ignores expected return (Heeswijk, 2012).

4.2 Real option analysis

Real option analysis is a method to value projects by modelling decisions in the option framework (Heeswijk, 2012). It is an extension of the standard option used to value financial assets (Luenberger, 1998). The owner of a standard option has the right instead of obligation to buy (call option) or sell (put option) the underlying assets at fixed price (exercise price). In real options, the underlying asset is the investment in tangible assets (Martínez-Ceseña and Mutale, 2011). The value of standard options and real option is determined by many factors like risk free rate, maturity, the value and volatility of underlying assets (Rumbauskaitė, 2011).

Real options cover the shortage of traditional value methods mentioned in section 4.1 (Triantis and Borison, 2001). Real options value the flexibility, of the owner to change their decision at the decision point to maximize profit or minimize loss based on the current information (Copeland and Keenan, 1988). Different from traditional value method, real options also adjust the discount rate to the risk profile (Heeswijk, 2012).

4.2.1 Mathematical Background

To begin with, we give the mathematical background to help to understand the real option analysis as is introduced by Dixit and Pindyck (1994).

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𝑑𝑧 = 𝜀𝑡√𝑑𝑡, (4.1)

where 𝜀_𝑡 is a random variable following normal distribution, with zero mean and unit standard deviation; dt is a time period which is infinitesimally small.

The simplest recapitulation of a Wiener process is the geometric Brownian motion (GBM) with drift:

dx = αxdt + σxdz, (4.2) where dx is the proportional change in variable x, α is the drift parameter (also called growth rate because it illustrates the trend of the variable), σ is the variance parameter, and dz is the increment of a Wiener process.

Ito’s lemma is used to determine the differentials of the functions of GBM. It can be regarded as a Taylor series expansion. Assume that x(t) follows GBM (as equation (4.2)), and consider a function F (x, t) that is differentiable more than twice in x and once in t. The differential of F (x, t) can be expressed as:

𝑑𝐹 = 𝐹′_{(𝑥)𝑑𝑥 + 𝐹}′_{(𝑡)𝑑𝑡 +}1 2𝐹

′′_{(𝑥)(𝑑𝑥)}2₊1 6𝐹

′′′_{(𝑥)(𝑑𝑥)}3_{+ ⋯. (4.3)}

Since E(dz) = 0, E[(dz)2_{] = dt, and ignore the terms approaching zero faster than dt, the}

dF can be expressed as:

𝑑𝐹 = 𝐹′_{(𝑥)𝑑𝑥 + 𝐹}′_{(𝑡)𝑑𝑡 +}1 2𝐹

′′_{(𝑥)(𝑑𝑥)}2_. _(4.4)

The Bellman equation is a necessary condition to get optimization in dynamic programming. Consider the investment in a firm value V (P), a function of its output price P which is a stochastic variable, the Bellman equation in discrete time is expressed as:

𝑉(𝑃_𝑡) = 𝐸[𝑃𝑡+1 1+𝑟+

1

1+𝑟𝑉(𝑃𝑡+1)], (4.5)

where V (Pt) is the firm value at time t. It equals the expectation of discounted cash flow

at time t+1 plus the continuation value after time t+1.

The Bellman principle can be used in continuous time, where the output price P follows GBM. The time interval dt is infinitesimally small. At next moment, time will move from 0 to dt, the firm will receive Pdt, the change of output price will be dP, and the value will change from V (P) to V (P+dP). Therefore the Bellman equation as follows:

𝑉 = 𝐸[𝑃𝑑𝑡 + e−𝑟𝑑𝑡_{𝑉(𝑃 + 𝑑𝑃)],} _(4.6)

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As described in Appendix A, the Bellman equation can be also expressed as: 𝑟𝑉 = 𝑃 + 𝛼𝑃𝑉′₊1

2𝜎2𝑃2𝑉′′, (4.7)

where r is the discount rate; α is the growth rate of the output price; σ is the standard deviation of the output price; and 𝑉′_and _𝑉′′_{are separately the first and second}

derivative of the value function. The first term of this equation on the right side can be the cash flow that the firm gets.

The hotelling rule shows that the percentage change in net price every unit of time should be equal to the discount rate to get the maximum present value of the resource capital during the extraction period. The condition can be expressed as

P’(t)/P(t)=r, (4.8) where r is the discount rate. This shows that the natural resources can be regarded as capital goods, meaning that the owner of the resources can get capital profit by preserving them. If the growth rate of the resource price equals to the the growth rate of other assets, the present value of exploiting the resource in the future will be equal to the value of extracting the same amount of the resource immediately. So, the value of the resource for the owner to extract it is the same as to preserve it.

4.2.2 Real option valuation method

Dixit and Pindyck (1994) give a frame working for real option valuation. The real option analysis is used to value a project by measuring its flows of payoffs and costs under specific uncertainties. An important assumption is that the uncertainties can be modelled by continuous time Ito’s process (Rumbauskaitė, 2011). The underlying asset of this option is the right which a company holds to invest in a project.

Assume that the value of the project is V(P) depends on the output price P of the project. And the price P follows GBM. So proportional changes in price P can be described mathematically as:

dP = μPdt + σPdz, (4.9) where dP is the change in P over an infinitesimally small interval of time, μ is the drift parameter, σ is the variance parameter, dz is the increment of a Wiener process. Then its expected value at time t equals (as showed in Appendix B):

E(P(t)) =𝑒𝜇𝑡_𝑃(0), _(4.10)

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Therefore the value of the project is: 𝑉(𝑃) = ∫ 𝑒∞ −𝑟𝑡_{𝐸(𝑃)𝑑𝑡} 0 = ∫ 𝑒(𝜇−𝑟)𝑡𝑃𝑑𝑡 = 𝑃 𝑟−𝜇 ∞ 0 . (4.11)

4.3 Trinomial tree model

According to Clewlow and Strickland (1998), the binomial model is a simple numerical method to value American style options and it can be generalized to trinomial tree model and extended into a grid. Under the assumption that it is a risk neutral market where all investors are risk neutral and then all assets earn risk free rate of interest, the equation (4.9) can be rewritten as:

dP = rPdt + σPdz, (4.12) where r is the risk free rate. If the asset pays at an annual dividend at rate δ, the drift parameter in equation (4.12) is subtracted by the dividend yield. Therefore the process can be expressed as:

dP = (r-δ)Pdt + σPdz. (4.13) The natural logarithm of the price (x = lnP) is commonly distributed with a constant mean and variance (Clewlow and Strickland, 1998). According to Ito’s lemma and the risk neutral consumption, the proportional change can be expressed as:

dx = νdt + σdz, (4.14) where

ν = r – δ – 1₂𝜎2_. _(4.15)

As showed in Figure 4.1, the price can grow or decrease by Δx, or stay the same, with the probabilities separately pu, pd, and pm in a small time interval Δt. Same as

Clewlow and Strickland (1998), the time interval can be chosen as:

Δx = σ√3Δ𝑡. (4.16)

Figure 4.1. Trinomial tree model of an asset price

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To get the relationship between the parameters, the following equation set is used:

E[Δx]= pu(Δx)+ pm(0)+ pd(-Δx)= νΔt (4.17)

E[Δx2_{]= p}

u(Δx2)+ pm(0)+ pd(Δx2)= σ2Δt+ν2Δt2 (4.18)

pu+ pm+ pd=1 (4.19)

Solving the equation set gives:

pu= 1₂(𝜎 2_Δ𝑡+𝜈2_Δ𝑡2 Δ𝑥2 + 𝜈Δ𝑡 Δ𝑥) (4.20) Pm= 1 −𝜎 2_Δ𝑡+𝜈2_Δ𝑡2 Δ𝑥2 (4.21) Pd= 1 2( 𝜎2_Δ𝑡+𝜈2_Δ𝑡2 Δ𝑥2 − 𝜈Δ𝑡 Δ𝑥) (4.22)

A trinomial tree can be formed by extending the trinomial process in Figure 4.1 and showed in Figure 4.2, where i is the step of time and j is the price level relative to the original price, meaning that at node (i, j) the time is t = iΔt, and the asset price Si, j = S

exp(jΔx). An option value at node (i, j) can be represented as Ci, j. The final time step N

is also the maturity T of the option. For a call option, the value of it at maturity is:

CN, j = max (0, SN, j – K), (4.23)

where K is the strike price.

Figure 4.2. A trinomial tree

Under the risk neutral assumption, the option values at forward nodes can be calculated as discounted expectations:

Ci, j = e-rΔt (pu Ci+1, j+1 + pm Ci+1, j + pd Ci+1, j-1) (4.24)

Consider at every time step i, the number of nodes is 2Nj + 1, Nj ≥ N rather than 2i

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Figure 4.3. A trinomial tree in a rectangular grid

The values of option at the top and bottom nodes at time i: 𝐶_𝑖,𝑁_𝑗 and 𝐶_𝑖,−𝑁_𝑗 need the option values at time i+1: 𝐶_𝑖+1,𝑁_𝑗₊₁ and 𝐶_{𝑖+1,−𝑁}_𝑗₋₁ which is unknown. So it is necessary to add boundary conditions to compute the grid (example for a European call option):

𝐶

𝑆 = 1 for S large (4.25)

𝐶

𝑆 = 0 for S small (4.26)

Using equations (4.23) and (4.24) in the term of grid, we get:

𝐶_{𝑖,𝑁𝑗}−𝐶_{𝑖,𝑁𝑗−1}

𝑆_{𝑖,𝑁𝑗}−𝑆_{𝑖,𝑁𝑗−1} = 1 for S large (4.27)

𝐶_{𝑖,−𝑁𝑗+1}−𝐶_{𝑖,−𝑁𝑗}

𝑆_{𝑖,−𝑁𝑗+1}−𝑆_{𝑖,−𝑁𝑗} = 0 for S small (4.28)

The explicit finite difference method can be obtained from the aforementioned model, which is used in this paper.

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5. Real option modelling in gas field depletion

Depicted in the previous section, the theory of real options is applied in this section to value investment in gas field depletion. As showed in section 3, a gas field has a limited and fixed quantity of natural gas. Therefore the investment problem is when to invest in the depletion of a gas field. The object of the option model is to find the optimal time for this investment. The uncertainty modeled as the stochastic variable is gas price. An important characteristic of the investment is the time elasticity since investors can change their investment decision based on new information about gas price and other market situations.

For the features mentioned above, real options valuation is an appropriate model to analyze this investment problem. The right of a company to invest in gas field depletion can be seen as a real option (Rumbauskaitė, 2011). And the company can choose to wait or invest immediately based on the gas price. The natural gas is the output of the project. If the gas price is high enough, it is optimal to invest the project. Otherwise, the investor should delay the investment. Therefore, the goal of this model is to estimate the threshold price P* at which it is optimal to invest in the gas field depletion.

5.1 One stage model

Consider a company has the right to invest in the project. And it has to pay the basic investment cost I which is a constant. The output of the project is natural gas, of which the price p is a stochastic variable following the geometric Brownian motion, meaning that:

dp = μpdt + σpdz, (5.1) where dp is the proportional change in variable p, μ is the drift parameter, σ is the

proportional variance parameter, and dz is the increment of a standard Wiener process. Assume that all natural gas in the gas field can be extracted and sold. Therefore the

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Suppose that the current level of gas price is too low for the investment. So the investor will wait until the gas price reach the trigger gas price p*. Therefore the investor pays cQ to get payoff pQ of the project. If the gas price is lower than p*, meaning that it is not optimal to invest immediately since the future cash flows (pQ -

cQ) are less than an option value while waiting V(p). To determine the trigger price p*,

it is necessary to estimate the value of the investment option.

The value of the investment option is dependent on the gas price p. According to the Bellman equation showed in section 4.2.1:

𝑟𝑉(𝑝) = 𝜇𝑝𝑉′_{(𝑝) +}1 2𝜎

2_𝑝2_𝑉′′_(𝑝), _(5.2)

where r is the discount rate; μ is the growth rate of the output price; σ is the standard deviation of the gas price; and 𝑉′_{(𝑝) and 𝑉}′′_{(𝑝) are separately the first and second}

derivative of the value function. The cash flow that the company gets is 0 since there is no payoff without investment on the gas field depletion. To begin with, guess the function of V(p) is:

𝑉(𝑝) = 𝐴𝑝𝛽_. _(5.3)

After substitution of V(p), the equation (5.2) becomes 𝑟𝑝𝛽 _{= 𝜇𝑝𝛽𝑝}𝛽−1₊1

2𝜎

2_𝑝2_{𝛽(𝛽 − 1)𝑝}𝛽−2_, _(5.4)

(𝜇 −1₂𝜎2_{) 𝛽 +}1

2𝜎2𝛽2− 𝑟 = 0. (5.5)

The solutions of equation (5.5) are: 𝛽₁ = −(𝜇− 1 2𝜎2)+√(𝜇− 1 2𝜎2) 2 +2𝜎2_𝑟2 𝜎2 ; (5.6) 𝛽₂ =−(𝜇− 1 2𝜎2)−√(𝜇− 1 2𝜎2) 2 +2𝜎2_𝑟2 𝜎2 . (5.7)

Additionally, β1 > 1, and β2 < 0. Therefore, the equation (5.3) becomes

𝑉(𝑝) = 𝐴𝑝𝛽1 + 𝐵𝑝𝛽2_. (5.8)

Consider that the value of the investment option will be worthless when the payoff of the project is almost nothing, meaning that the value of the option is approach to 0 if the gas price approaches 0. Thus the boundary condition is

𝑉(0) = 0. (5.9)

The term 𝐵𝑝𝛽2_{in equation (5.8) is approach to infinite when p approaches 0. Hence}

the term B is set to 0 and the solution becomes

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When the gas price is p*, the company will immediately invest in the project. And the value of the investment option equals to the net payoff of the project. This is called the “value matching” condition (Dixit and Pindyck, 1994; Murto and Nese, 2003). In addition, optimality requires that even the derivatives should be the same at the transition point, meaning that the slopes of option value function and net payoff function should match. This is called the “smooth-pasting” condition. Therefore, there are two boundary conditions:

𝑉(𝑝∗_{) = 𝑝}∗_{𝑄 − 𝑐𝑄} _(5.11)

𝑉′(𝑝∗_{) = (𝑝}∗_{𝑄 − 𝑐𝑄)′.} _(5.12)

It follows that:

𝐴𝑝𝛽1 = 𝑝𝑄 − 𝑐𝑄 (5.13)

𝛽1𝐴𝑝𝛽1−1 = 𝑄. (5.14)

The solutions of the equations above are: 𝑝∗ ₌ 𝛽1 𝛽₁− 1𝑐 (5.15) 𝐴 = 𝑄 𝛽₁( 𝛽₁ 𝛽₁− 1𝑐)(1−𝛽1) (5.16)

Then the optimal investment threshold 𝑝∗_{is determined. It is dependent on the}

coefficient 𝛽₁ and the production cost c. If the production cost increases (with other parameters permanent), the investor would like to wait longer and pay more for the investment. And the coefficient 𝛽1 depends on the growth rate, standard deviation of

gas price, and the discount rate. The larger volatility σ, the closer 𝛽₁ to 1, leading to higher trigger price 𝑝∗_.

5.2 Two stages model

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value at the second stage, then works backwards to estimate the value at the first stage and determine the optimal timing of the investment. The decision process is shown in figure 5.1.

Figure 5.1. The investment and producing process

The explicit finite difference method based on trinomial tree method is used to model this two stage depletion, which will be proved to be able to get the same result as the traditional real option later. Clewlow and Strickland (1998) apply explicit finite difference method to the Black-Scholes Partial Differential Equation (PDE) and get

−𝐹 𝑡 = 1 2𝑃2𝜎2 2𝐹 𝑃2+ (𝑟 − 𝛿)𝑆 𝐹 𝑃− 𝑟𝐹, (5.17)

where F is the value of a derivative, which is dependent on the asset price P (following the GBM) at time t; σ is the volatility parameter; r is the discount rate; δ is the dividend yield paid by the asset. All of the European and American options of which the payoff is dependent on only one asset following GBM are regulated by this PDE. Suppose that

x=ln(P), the equation (5.18) becomes

−𝐹 𝑡 = 1 2𝜎2 2𝐹 𝑥2+ 𝜈 𝐹 𝑥− 𝑟𝐹, (5.18)

where ν is the same as in equation (4.15). Similar to the trinomial tree method, the time and space in this model is also divided up into discrete intervals as Δt and Δx. The equation (5.19) is approximated by a forward difference for _𝐹_𝑡 and central differences for _2_𝑥𝐹₂ and _𝐹_𝑥 (Clewlow and Strickland, 1998) and becomes

−𝐹𝑖+1,𝑗−𝐹𝑖,𝑗 Δ𝑡 = 1 2𝜎2 𝐹 𝑖+1,𝑗+1−2𝐹𝑖+1,𝑗+𝐹𝑖+1,𝑗−1 Δ𝑥2 + 𝜈 𝐹𝑖+1,𝑗+1−𝐹𝑖+1,𝑗−1 2Δ𝑥 − 𝑟𝐹𝑖+1,𝑗. (5.19)

The equation (5.20) can be rewritten as follows:

𝐹_𝑖,𝑗= 𝑝_𝑢𝐹_{𝑖+1,𝑗+1}+ 𝑝_𝑚𝐹_𝑖+1,𝑗+ 𝑝_𝑑𝐹_{𝑖+1,𝑗−1} (5.20) Stage 1 Stage 2 Waiting American Call Producing European Call Waiting American Call Producing European Call T2 T1 Second payoff: 𝑉2

Invest when the price is 𝑝₂∗ Invest when the

price is 𝑝₁∗

Second option value: 𝐹2

First payoff: 𝑉₁

A B C D E

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20 𝑝𝑢 = 𝛥𝑡( 𝜎2 2Δ𝑥2 + 𝜈 2Δ𝑥) (5.21) 𝑝𝑚 = 1 − 𝛥𝑡 𝜎2 Δ𝑥2− 𝑟𝛥𝑡 (5.22) 𝑝𝑑 = 𝛥𝑡( 𝜎2 2Δ𝑥2 − 𝜈 2Δ𝑥) (5.23)

Now consider the producing period at stage 2. The company decides to invest in the project of when the gas price is p2 at time D in Figure 1, and get the payoff at the end

of the producing time (point E), meaning that the expected payoff at the start of producing process (at time E) is

𝑉₂ = (𝑝₂𝑒𝜇𝑇2∗ 𝑄)𝑒−𝑟𝑇2 = 𝑝₂𝑒−(𝑟−𝜇)𝑇2 ∗ 𝑄, (5.24)

where μ is the growth rate of the gas price, r is the discount rate, and T2 is the production

time. The value of this project in producing process can be regarded as a European call option since the time period of producing natural gas is fixed. By using the explicit finite difference method, we can get an array of the expected return V2(p) of producing

natural gas. This process can be realized through Python with which the code is showed in Appendix C.

Figure 5.2. The expected return at the second stage

Figure 5.2 illustrates the array of the expected payoff with different price of natural gas. The parameter used are p2=10, c=5, Q=10, T2=1, σ=0.2, r=0.06, δ=0.03. It can be

seen that the expected payoff increases with the growth of gas price. This line also looks similar to the payoff line of a European option. Based on the same parameter, the result of equation (5.24) is 94.04, which equals to the midpoint of the array of the expected return (represented as V2(0,0) in the Appendix C).

Since the investor has the right to choose either waiting or investing immediately, the waiting time can be infinite if the price is always below the threshold. Therefore, the

V2(p)

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value of this project in waiting time can be regarded as an American call option with a significantly long maturity. The option value at maturity for node (N, j) is

𝐹2(𝑁, 𝑗) = max⁡(0, ⁡ 𝑉2(0, 𝑗) − 𝑐𝑄). (5.25)

As showed in equation (5.25), the strike price of this American call option is the investment cost at the second stage cQ. Another difference from a European call option is that an American option allows the option holder to exercise it at any time before or including the maturity date, while a European option can only be exercised at maturity. Thus, the early exercise test is necessary. At node (i, j), the early exercise condition is

𝐹2(𝑖, 𝑗) = max⁡(𝐹2(𝑖, 𝑗), ⁡ 𝑉2(0, 𝑗) − 𝑐𝑄). (5.26)

The code for this American call option is showed in Appendix C, including the check with the result of real options. The equation (5.24) shows the present payoff at the start of the producing (time D). The two boundary conditions are

𝐴𝑃₂𝛽1 _{= 𝑃}

2𝑒−(𝑟−𝜇)𝑇2𝑄 − 𝑐𝑄 (5.27)

𝛽₁𝐴𝑃₂𝛽1−1_{= 𝑄𝑒}−(𝑟−𝜇)𝑇2. (5.28)

Using the same parameter as in the producing process, and setting the maturity of the American call option is fifty years, we find that the American call option value F2(0,0)

is 51.97 and the real option value C2 is 52.04. The plots of both kind of option value

with the payoff is similar (shown in Figure 5.3). The threshold price at the second stage is 15.46 for American call option and 15.50 for real option.

Figure 5.3. The American call option and real option value in the second stage

Now we work backwards to estimate the value at the first stage by applying the explicit finite difference method. Similar to the second stage, the producing process is a European option. The value of this European call option at the maturity (time C) is

𝑉1(𝑁, 𝑗) = 𝐹2(0, 𝑗) + 𝑝1(𝑁, 𝑗)𝑄, (5.29)

where F2 is the American option value at stage 2, p1 is the gas price at the first stage, p2

F2(p)

p2

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22

and T1 is the producing period of the first stage. To estimate the option value of the

waiting time, the American call option by explicit finite difference is applied, which is the same as at the second stage. In this way, the trigger price for the first stage is determined.

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6. Numerical example

The abstract model showed in the section 5 gives the method to solve the investment problem. In this section, the model will be applied in a numerical case based on the gas market in the Netherlands. To begin with, the actual value of the parameters will be given. Then the threshold price will determined by using the model in section 5.

6.1 The value of the parameters

The most important parameters should be provided are the coefficients associated with gas price, the production cost, and the discount rate. The gas price p is settled as the uncertainty of the model and follows the GBM with the drift parameter μ and volatility parameter σ. According to IEA, the price of National Balancing Point (NBP) and Title Transfer Facility (TTF) was USD 5 per million British thermal unites (MBtu) in May 2017. And they predict that the forward curves are higher around USD 5.2/MBtu in 2018-2019 and are forecasted to be stable through 2022. The drift coefficient of the gas price can be estimated as the compound annual growth rate (Rumbauskaitė, 2011). Therefore the growth rate is assumed as μ = 0.0079. To calculate volatility of gas price, 1-year data of Henry Hub Natural Gas Spot Price (USD/MBtu) is obtained from the website of U.S. Energy Information Administration (EIA). The volatility is σ = 0.42.

Furthermore, the Weighted Average Cost of Capital (WACC) is a frequently used discount rate. The WACC of a vertically integrated energy firm operating in Great Britain estimated by the Competition & Markets Authority (CMA) is between 7.7 and 9.5%. Thus we can assume that r = 8.5%. As showed in equation (4.12), the dividend yield of the gas depletion is δ = r – μ = 7.71%. The model from Mistré, Crénes, and Hafner (2018) shows the gas production cost ranges from $2.5/MBtu to $6.0/MBtu (equal to €0.77/m3_{and €1.86/m}3₎2_{. Therefore the assumption of gas production cost can}

be €1.10/m3_.

The data from Markets Insider shows that the gas prices (Henry Hub) changed between $2.53/MBtu to $3.66/MBtu (equal to €0.78/m3_{and €1.13/m}3_{) in the last 52}

2_{1 MBtu = 2.78 m}3_{, 1 dollar = 0.86 euro (average between 25 June and 25 July, 2018). Retrieved from}

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weeks. Hence we assume that the gas price is €0.85/m3_{. According to NLOG, the total}

estimated reserve of Dutch gas field is 1600 billion m3_{. Therefore, we assume that the}

output at every stage is 800 billion m3_{. Additionally, we assume that the time of}

producing (the maturity of the European option) is 10 year. It is reasonable to assume that the waiting time at both stages is sixty years since the option lasts forever, and the time of sixty years is long enough to be considered as perpetuity. All the parameters are showed in table 6.1.

Table 6.1. Numerical results of the second stage

Growth rate of gas price μ 0.79%

Volatility of gas price σ 0.42

Discount rate r 8.50%

Dividend yield δ 7.71%

Production cost, €/m3 _c _1.10

Gas price, €/m3 _p _0.85

Output of each stage, billion m3 _Q

1,Q2 800

The time of producing at each stage, year T1,T2 10

Waiting time at each stage, year t1, t2 60

6.2 The numerical results

Given the value of parameters in section 6.1, we can solve the investment problem to find the trigger price of natural gas for the investment of gas field depletion. The numerical results in the two stages are showed in table 6.2 and 6.3.

Table 6.2. Numerical results of the second stage

The expected payoff in the second stage: V2 (billion €) 314.53

The real option value in the second stage: F2 (billion €) 66.77

The trigger price in the second stage: P2* (€/m3) 6.80

As showed in table 6.2, the threshold of investment in stage 2 is €6.80/m3_{, which is}

much higher than the current gas price (€0.85/m3_{). Therefore, the price of natural gas}

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Table 6.3. Numerical results of the first stage

The expected payoff in the first stage: V1 (billion €) 374.77

The real option value in the first stage: F1 (billion €) 105.47

The trigger price in the first stage: P1* (€/m3) 5.53

Table 6.2 illustrates the numerical results of stage 1. The trigger price at the first stage is €5.53/m3_{, which is lower than that at the second stage, but still higher than the current}

price of natural gas. Hence, the investor is not willing to invest in the project of gas field depletion immediately and tends to keep waiting until the expected payoff in the first stage reaches 374.77 billion euro.

Figure 6.1 shows the option value and the profit at the second stage. The blue line describes the option value of the second stage, and the red line shows the profit that the investor gets if he chooses to invest. The junction point is the threshold at the second stage. The curve of real option value 𝐴2𝑝2𝛽2 is the same as the blue line in figure 6.1.

The red line is a linear line which is matched with the profit equation: 𝑝2𝑒−(𝑟−𝜇)𝑇2𝑄 −

𝑐𝑄. When the gas price is lower than trigger price p2*, the blue line is above the red line,

meaning that the option value is higher than the profit of immediate investment. Therefore the investor tends to wait. If the gas price reaches the threshold, the blue and red line overlaps, which means that the value of option equals the profit. Thus the investor decides to make the investment. This figure also explains the “smooth-pasting” and “value-matching” condition showed in equation (5.11) and (5.12).

Figure 6.1. The option value and the profit at the second stage

The option value and the profit at the first stage is showed in figure 6.2. Same as in

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figure 6.1, the blue line describes the option value owned by the investor in waiting time, while the red one is the profit of the first investment. The intersection point is the threshold of the first stage. These two figures are similar, but the main difference is that the red line in figure 6.2 is nonlinear. Because the profit of the first investment is 𝑝1𝑒−(𝑟−𝜇)𝑇1𝑄 − 𝑐𝑄 plus the expected value of the second investment, which is

nonlinear. When the gas price is lower than the first trigger price p1*, the blue line

dominates, meaning that the investor tends to hold the option instead of invest immediately. When the gas price reaches the threshold, these two line overlaps again. The investor would like to make the investment.

Figure 6.2. The option value and the profit at the first stage

Note that the second trigger price (€6.80/m3_{) is higher than the first one (€5.53/m}3_),

which is consistent with our instinct. As discussed before, due to the finite resource of natural gas, it is necessary to carefully decide the optimal path of gas depletion. At start of the second waiting time, the half of gas reserve has already been explored in the first stage. Only the last half is available. In this case, the further depletion could be decided more careful. Therefore it is reasonable to set a higher threshold for the second investment than the first one.

6.3 Sensitivity analysis

After showing the specific numerical case in section 6.2, we examine the effects of the main parameters (volatility of gas price, the growth rate of gas price, and discount rate) on the trigger prices at two stages in this subsection.

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Firstly, we analyze the impact of the volatility which is used to describe the degree of uncertainty. To examine this impact, the volatility of gas price is ranged from 0.1 to 0.6. The corresponding threshold at two stages is calculated and showed in figure 6.3.

Figure 6.3 Impact of volatility on the trigger price

As showed in figure 6.3, the trigger price p2* at stage 2 is always higher than the that

(p1*) at stage 1, regardless of the volatility. Additionally, both blue and orange lines

show an upward trend, meaning that the trigger prices at both stages increase when the uncertainty over the gas price increases. This implies that, when facing more uncertainties about the future cash flow, the option owned by the investor is more valuable. Therefore the investor is willing to wait longer. Murto and Nese (2003) also conclude that the uncertainty can delay the investment.

Furthermore, the growth rate of gas price affects the value of β1 in equation (5.6),

hence influences the trigger price. The impact of growth rate on the trigger gas prices is described in figure 6.4.

The growth rate ranges from 0.005 to 0.075. Because of equation δ = r – μ (r = 0.085), the dividend yield changes from 0.08 to 0.01. Same as the result in figure 6.3, the second threshold p2* is always higher than the first one p1*. Both trigger prices increase with

the increase of gas price growth rate, and the change of thresholds accelerates. For example, when the growth rate changes from 0.025 to 0.035, p2* increases from 7.0 to

7.3. When the growth rate shift from 0.065 to 0.075, the growth of p2* is 21.7 – 12.4 =

9.3. This means that investors tend to postpone more the investment with higher growth rate of gas price. It also illustrates the negative impact of the dividend yield on trigger

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prices.

Figure 6.4 Impact of growth rate on the trigger price

Another analysis is about the impact of the discount rate on trigger prices. This effect also comes from the impact on the value of β1 in equation (5.6).

Figure 6.5 Impact of discount rate on the trigger price

The discount rate ranges from 0.05 to 0.35. Figure 6.5 illustrates that the discount rate r has a positive impact on the trigger prices at both stages. The second threshold is again always higher than the first one, no matter how discount rate changes. Note that the change of the second threshold keeps similar with the fixed increment of the discount rate, which is different from the situation of growth rate. Therefore we can conclude that a higher discount rate encourages the investor to postpone the investment.

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7. Conclusions

In this paper, we focus on the problem of a finite resource - natural gas and aim to find the optimal depletion path. The main research problem of this paper is to determine the optimal timing of investment in gas field depletion. The only uncertainty of this model is assumed to be the gas price. The investment process is divided into two stages that are modeled as American call option following with the European call option. The real option modeling illustrates that the threshold of the investment depends on various parameters such as the investment costs, the discount rate, and the growth rate of the gas price, and is independent on the price of natural gas.

To apply this model in reality, the data from the gas market in the Netherlands are used as a numerical example. The result shows that the estimated threshold for investment in gas field depletion at the second stage is higher than that at stage 1, meaning that the investor will wait longer to invest at the second stage. Because the investor has to consider the limited gas reserve and carefully decide the further depletion. The length of this waiting period is dependent on the current price of natural gas.

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research. There are a number of companies in the market, and the competition will reduce the profit of the investment in the project, leading a longer waiting period. Furthermore, the investment process is only divided into two stages. However, it is more realistic that a gas field is depleted in more than two steps. The solving process will be complicated with a number of stages to estimate the same amount of trigger price.

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Appendix A

The Bellman equation in continuous time is:

𝑉 = 𝐸[𝑃𝑑𝑡 + e−𝑟𝑑𝑡_{𝑉(𝑃 + 𝑑𝑃)].}

For 𝑉(𝑃 + 𝑑𝑃), the function is:

𝑓(𝑥 + 𝑑𝑥) = 𝑓(𝑥) +𝑑𝑓_𝑑𝑥𝑑𝑥. Therefore we can get:

𝑉(𝑃 + 𝑑𝑃)= 𝑉(𝑃) + 𝑉′_{(𝑃)𝑑𝑃 +}1 2𝜎2𝑃2𝑉′′(𝑃)𝑑𝑡 = 𝑉(𝑃) + 𝜇𝑃𝑉′_{(𝑃)𝑑𝑡 + 𝜎𝑃𝑉}′_{(𝑃)𝑑𝑧 +}1 2𝜎2𝑃2𝑉′′(𝑃)𝑑𝑡 (A.1) 𝑒−𝑟𝑡_{𝑉(𝑃 + 𝑑𝑃)= (1 − 𝑟𝑑𝑡)(𝑉(𝑃) + 𝜇𝑃𝑉}′_{(𝑃)𝑑𝑡 + 𝜎𝑃𝑉}′_{(𝑃)𝑑𝑧 +}1 2𝜎2𝑃2𝑉′′(𝑃)𝑑𝑡) = 𝑉 + (−𝑟𝑉 + ⁡ 𝜇𝑃𝑉′_{(𝑃) +}1 2𝜎 2_𝑃2_𝑉′′_{(𝑃)) 𝑑𝑡 + ⁡ 𝜎𝑃𝑉}′_{(𝑃)𝑑𝑧. (A.2)}

So the Bellman equation:

𝑉= 𝐸[𝑃𝑑𝑡 + e−𝑟𝑑𝑡_{𝑉(𝑃 + 𝑑𝑃)]}

= 𝐸 [𝑉 + (P − 𝑟𝑉 + ⁡ 𝜇𝑃𝑉′_{(𝑃) +}1

2𝜎2𝑃2𝑉′′(𝑃)) 𝑑𝑡 + 𝜎𝑃𝑉′(𝑃)𝑑𝑧] = 𝑉 + (P − 𝑟𝑉 + ⁡ 𝜇𝑃𝑉′_{(𝑃) +}1

2𝜎2𝑃2𝑉′′(𝑃)) 𝑑𝑡. (A.3)

Then we can get the Bellman equation:

𝑟𝑉 = 𝑃 + 𝜇𝑃𝑉′₊1 2𝜎

2_𝑃2_𝑉′′_. _(4.5)

Appendix B

Hull (2015) shows the equation deducing to get equation (4.10). Assume that the value of a variable x follows the Ito process:

𝑑𝑥 = 𝑎(𝑥, 𝑡)𝑑𝑡 + 𝑏(𝑥, 𝑡)𝑑𝑧, (B.1)

where a(x,t) and b(x,t) are functions of x and t; dz is a Wiener process. The drift rate of

x is a, and the variance rate of x is b2_{. Consider a function G of x and t, according to}

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32 𝑑𝐺 = (𝐺 𝑥𝑎 + 𝐺 𝑡 + 1 2 2_𝐺 𝑥2𝑏2) 𝑑𝑡 + 𝐺 𝑥𝑏𝑑𝑧, (B.2)

where dz is also a Wiener process same with that in equation (B.1). Therefore, the function G follows the Ito process.

Suppose the function G is:

𝐺 = ln⁡(𝑥), (B.3)

the equation (B.2) can be expressed as: 𝑑(𝑙𝑛𝑥) =1 𝑥𝑑𝑥 − 1 2 1 𝑥2𝜎2𝑑z2. (B.4)

Substitute equation (B.1) to dx, we can get:

𝑑(𝑙𝑛𝑥) = 𝜇𝑑𝑡 + 𝜎𝑑𝑧 −1 2

1

𝑥2𝜎2𝑑z2. (B.5)

Calculate the integration of x, we can get the equation as follows: 𝑙𝑛𝑥 − 𝑙𝑛𝑥(0) = (𝜇 −1

2𝜎2) 𝑡 + 𝜎𝑧. (B.6)

Therefore, the variable x can be expressed as:

𝑥 = 𝑥(0) ∗ 𝑒(𝜇−12𝜎2)𝑡+𝜎𝑧. (B.7)

So the expectation of x:

𝐸(𝑥) = 𝐸(𝑥(0) ∗ 𝑒(𝜇−12𝜎2)𝑡+𝜎𝑧_). (B.8)

If a variance x follows the normal distribution with mean equaling μ and variance being

σ2_{, the following equation is always satisfied:}

𝐸(e𝑥_{) = 𝐸(𝑒}(𝜇+1_2𝜎2)_). (B.9)

Apply equation (B.9) in equation (B.8), we can get this equation:

𝐸(𝑥) = 𝑥(0)𝑒𝜇𝑡_, _(B.10)

which is the same as equation (4.10).

Appendix C

The code used on Python is shown as follows:

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# Stage 2: producing period (time D to E), similar to the European call option # Assume to get payoff once at the end of the producing (time E and C) # initial parameters

p2=10 # the price at the start of producing in stage 2 c=5 # the production cost

Q=10 # the output

T2 = 1 # the second producing period sig = 0.2 # the volatility of gas price r = 0.06 # the discount rate div = 0.03 # the dividend yield N = 1000 # the time step

Nj = 1000 # the number of nodes at every time step # indirect parameters

dt = T2/N # the time internal dx = sig*sqrt(3*dt) # the space step

nu = r - div -0.5*(sig**2) # the drift parameter edx = exp(dx)

pu = (1/2)*dt*((sig/dx)**2 + nu/dx) # the probability that the price grow pm = 1 - dt*(sig/dx)**2 - r*dt # the probability that the price do not change pd = (1/2)*dt*((sig/dx)**2 - nu/dx) # the probability that the price decrease # initialise option values at maturity

p2t=[0]*(2*Nj+1) # the gas price at the beginning of second producing period

V2 = np.zeros(( 2*Nj+1 , 2*Nj+1) , dtype=np.float64) # the European call option in stage 2 p2t[-Nj]=p2*exp(-(Nj)*dx) # the gas price at node (N.-Nj)

for j in range(-Nj+1,Nj+1): # calculate the other price in time step N p2t[j]=p2t[j-1]*edx

for j in range(-Nj,Nj+1): # the call option value is equal to the positive payoff or 0 V2[N,j]=p2t[j]*Q

# step back for i in range(N-1,-1,-1): for j in range(-Nj+1,Nj):

V2[i,j]=(pu*V2[i+1,j+1]+pm*V2[i+1,j]+pd*V2[i+1,j-1]) # already discounted # boundary conditions

V2[i,-Nj] = V2[i,-Nj+1]

V2[i,Nj] = V2[i,Nj-1]+(p2t[Nj]-p2t[Nj-1])*Q

Expected_payoff2 = V2[0,0] # the value in the middle of the grid print('The expected payoff in the second stage:')

print(Expected_payoff2) # for check

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34 mu = r-div # the growth rate of gas price

Pcheck = p2*Q*exp(-(r-mu)*T2) # dicount to the start of producting period print('Check the expected payoff in the second stage:')

print(Pcheck)

# the results are correct

# graph of the expected payoff in the second stage p2t_array=np.array(p2t) V2_array=np.array(V2[0,:]) fig, ax =plt.subplots() ax.plot(p2t_array, V2_array, 'b.') ax.set_xlim([0,20]) ax.set_ylim([-50,200])

# Stage 2: waiting time (time C to D), similar to American call option # step back: American call

t2=50 # this option will not be exercised if investor keep waiting, so the maturity is very long N2=N*(t2//T2) # make sure that the dt and dx in every regime is the same

dt=t2/N2 dx = sig*sqrt(3*dt) nu = r - div -0.5*(sig**2) edx = exp(dx) pu = (1/2)*dt*((sig/dx)**2 + nu/dx) pm = 1 - dt*(sig/dx)**2 - r*dt pd = (1/2)*dt*((sig/dx)**2 - nu/dx)

F2 = np.zeros(( 2*N2+1 , 2*Nj+1) , dtype=np.float64) # the option value of the second stage # initialise option values at maturity

for j in range(-Nj,Nj+1):

F2[N2,j]=np.maximum(0,V2[0,j]-c*Q) # the profit of this investment when the investor choose to invest # step back for i in range(N2-1,-1,-1): for j in range(-Nj+1,Nj): F2[i,j]=(pu*F2[i+1,j+1]+pm*F2[i+1,j]+pd*F2[i+1,j-1]) # boundary conditions F2[i,-Nj] = F2[i,-Nj+1] F2[i,Nj] = F2[i,Nj-1]+(V2[0,Nj]-V2[0,Nj-1]) # apply early exercise condition

F2[i,j]=np.maximum(F2[i,j],V2[0,j]-c*Q) # excercise earlier when the value is higher than the profit American_call2 = F2[0,0] # the value in the middle of the grid

print('The american option F2(0,0) in the second stage:') print(American_call2)

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p2=10 c=5 Q=10 sigma = 0.2 r = 0.06 div = 0.03 T2 = 1 # indirect parameters mu = r-div beta1 = (-(mu-(1/2)*sigma**2)/sigma**2)+(sqrt((mu-(1/2)*sigma**2)**2+2*r*sigma**2))/sigma**2 p2_star = ((beta1)/(beta1-1))*c*exp((r-mu)*T2) A = Q*(exp(-(r-mu)*T2))/(beta1*(p2_star**(beta1-1))) if p2 > p2_star: F = p2*Q*exp(-(r-mu)*T2)-c*Q else: F = A*(p2**beta1)

print('For check: the real option value in the second stage:') print(F)

print('The threshold price in the second stage:') print(p2_star)

# find the threshold price in the second stage p2t_array=np.array(p2t)

F2_array=np.array(F2[0,:]) V2_array=np.array(V2[0,:]-c*Q)

min2=np.min(p2t_array[F2_array==V2_array]) # the minimum price at which the option value equals to profit max2=np.max(p2t_array[F2_array>V2_array]) # the maximum price at which the option value is higher than profit print('The threshold price in the second stage:')

print(min2) # this result is more close to that with real option method, so we choose to use this one print(max2)

# the trigger prices calculated by these three methods are very closed to each other # graph of the real option value

Fcheck=A*p2t_array**beta1 fig, ax =plt.subplots() ax.plot(p2t_array, Fcheck, 'b.') ax.set_xlim([0,20])

ax.set_ylim([-50,200])

# graph of the american option value in the second stage fig, ax =plt.subplots()

ax.plot(p2t_array, F2_array, 'r.') ax.set_xlim([0,20])

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# Now we get an array of option value at stage 2: F2 (at time C), then we work backwards to stage 1

# stage 1:producing period (time B to C), also similar to European call option p1=10 # the price at point B

c=5 Q=10

T1=1 # the producing period at the first stage sig = 0.2 r = 0.06 div = 0.03 N=1000 Nj=1000 dt=T1/N dx = sig*sqrt(3*dt) nu = r - div -0.5*(sig**2) edx = exp(dx) pu = (1/2)*dt*((sig/dx)**2 + nu/dx) pm = 1 - dt*(sig/dx)**2 - r*dt pd = (1/2)*dt*((sig/dx)**2 - nu/dx) #initialise option values at maturity

p1t = [0]*(2*Nj+1) # the gas price at the start of first producing period (time B)

V1 = np.zeros(( 2*Nj+1 , 2*Nj+1) , dtype=np.float64) # the European call option value at stage 1 p1t[-Nj]= p1*exp(-(Nj)*dx) # the calcuation is the same as that in the first stage

for j in range(-Nj+1,Nj+1): p1t[j]=p1t[j-1]*edx

#initialise option values at maturity for j in range(-Nj,Nj+1): V1[N,j]=F2[0,j]+p1t[j]*Q # step back for i in range(N-1,-1,-1): for j in range(-Nj+1,Nj): V1[i,j]=(pu*V1[i+1,j+1]+pm*V1[i+1,j]+pd*V1[i+1,j-1]) # boundary conditions V1[i,-Nj] = V1[i,-Nj+1] V1[i,Nj] = V1[i,Nj-1]+(p1t[Nj]-p1t[Nj-1])*Q

Expected_payoff1 = V1[0,0] # the expected payoff at stage 1 print('The expected payoff in the first stage:')

print(Expected_payoff1)

#stage 1: waiting time (time A to B), also similar to American call

t1=50 # this option will not be exercised if investor keep waiting, so the maturity is very long N1=N*(t1//T1)

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dx = sig*sqrt(3*dt) nu = r - div -0.5*(sig**2) edx = exp(dx) pu = (1/2)*dt*((sig/dx)**2 + nu/dx) pm = 1 - dt*(sig/dx)**2 - r*dt pd = (1/2)*dt*((sig/dx)**2 - nu/dx)

F1=np.zeros(( 2*N1+1 , 2*Nj+1) , dtype=np.float64) # the option value of the first stage #initialise option values at maturity

for j in range(-Nj,Nj+1): F1[N1,j]=np.maximum(V1[0,j]-c*Q,0) # step back for i in range(N1-1,-1,-1): for j in range(-Nj+1,Nj): F1[i,j]=(pu*F1[i+1,j+1]+pm*F1[i+1,j]+pd*F1[i+1,j-1]) # boundary conditions F1[i,-Nj] = F1[i,-Nj+1] F1[i,Nj] = F1[i,Nj-1]+(V1[0,Nj]-V1[0,Nj-1]) #apply early exercise condition

F1[i,j]=np.maximum(F1[i,j],V1[0,j]-c*Q)

American_call1 = F1[0,0] # the option value of the first stage print('The american call option value in the first stage:') print(American_call1)

#plotting a graph p1t_array=np.array(p1t) F1_array=np.array(F1[0,:]) V1_array=np.array(V1[0,:]-c*Q)

fig, ax =plt.subplots() #graph of stage 1 ax.plot(p1t_array, F1_array, 'b.')

ax.plot(p1t_array,V1_array, 'r.') ax.set_xlim([0,30])

ax.set_ylim([-50,400])

print('The threshold price in the first stage:') min1=np.min(p1t_array[F1_array==V1_array]) print(min1)

max1=np.max(p1t_array[F1_array>V1_array]) print(max1)

fig, ax =plt.subplots() #graph of stage 2 ax.plot(p2t_array, F2_array, 'b.')

ax.plot(p2t_array,V2_array, 'r.') ax.set_xlim([0,30])

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Investment in Gas Field Depletion