a real-option approach for portfolio management of upstream hydrocarbon...

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A Real-Option Approach for Portfolio Management of Upstream Hydrocarbon Assets

Bacel Maddah, Ali Yassine, Imad G. Melki

Engineering Management Program, American University of Beirut, P.O. Box 11-0236, Riad El Solh, Beirut 1107 2020, Lebanon

Abstract. We develop an optimization-based decision support system (DSS) that assists oil companies in upstream exploration and production decisions for a portfolio of hydrocarbon assets over time. Specifically, the DSS is based on dynamic programming (DP) and utilizes a probabilistic hydrocarbon price (geometric Brownian motion) and updates the probability distribution of asset reserves across the portfolio based on the outcome of ongoing explorations. The DSS can be seen as valuating a series of two-asset real options, with the underlying assets being the hydrocarbon reserve and price. Given possible realizations of price and reserve, the DSS proposes optimal exploration and development scenarios. The challenge in developing these scenarios is high computer storage. However, we mitigate this by establishing structural properties of the production decisions. Numerical analysis indicates that exploration decisions are greatly affected by the correlation of reserves in different hydrocarbon assets and the volatility of the hydrocarbon price. High correlation between assets leads to successive exploration decisions and high volatility shifts decisions to later periods to benefit from further price fluctuations.

Keywords: hydrocarbon (oil and gas), DSS, dynamic programming, real options, exploration and production decisions

1 Introduction

Oil and gas exploration and production is a risky business with high potential

rewards. The hydrocarbon exploration frontier has moved to deep offshore water where

drilling and production have a high cost and require cutting edge technology and

expertise. Additionally, success in these projects is low and decisions taken incur

millions of dollars of investment and have long-term (e.g., 20- to 30-year) ramifications.

In this high risk / high reward environment, the proper management of portfolio

of O&G assets is essential for an oil company to operate profitably and for a

government to effectively exploit its natural resources. Thus, there is a significant

importance in developing optimization models for planning the oil and gas exploration

and production industry which capture all the complex trade-offs. However, the

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development of such models is riddled with many complications and challenging factor,

which include,

High uncertainty in critical factors, most notably, the amount of hydrocarbon in

the reserve and the future price,

Complex processes of exploration, development and production stages with

extremely high capital and operating costs and, in many countries, a turmoil of

political and security conditions,

Conflicting interests between the oil company typically baring most of the

exploration and production risk and the government trying to maximize its

benefits from depleting natural resources.

The reserve portfolio effect reflecting the dependency among reserve

uncertainties, in the sense that a commercial discoveries (or lack of) in one asset

affect reserve estimates in other assets.

Coping with uncertainties of price and reserve and price is perhaps the major

challenge in O&G development. On the one hand, prior to exploration drilling the

reserve level is highly uncertain. The existence of such reserves is indicated by

geological formations revealed by seismic studies, which does not guarantee

commercial quantities. After exploration drilling, most of the uncertainty in reserve

is resolved. However, uncertainty persists, since as it is well-known, one cannot

figure out the full reserve potential before the last drop of it is exploited. On the

other hand, complex factors of supply and demand (weather, political conflicts,

wars, technology change, etc.) affect the price of hydrocarbon, which leads to high

volatility and continuous fluctuations. Coupled with high costs that must be covered

by abundant revenues, price volatility has an amplified effect.

Therefore, it is important to capture the effect of high uncertainties and the

complex development factors in a decision support system (that assists in

exploration and production decisions) in upstream O&G. In this paper, we develop

such DSS that attempts to capture effectively the relevant factors. Our DSS is based

on stochastic dynamic programming which stipulates appropriate exploration,

development and production decisions for all possible future scenarios of price and

reserve. It has the following distinctive features:

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Capturing the uncertainty in reserve through appropriate probabilities

distribution, with correlation factors adjustable based on exploration outcome.

This allows learning from sequential drilling.

Capturing the random price fluctuation over time via an appropriate, geometric

Brownian motion model. This price model, common in financial engineering, is

well-known to account for complex forces of supply and demand (e.g.

Lunenberger 1998).

Adopting a reasonable hydrocarbon cost structure with high capital (capex) and

appropriate operational costs (opex) and with realistic factors such as the

deliverability which restricts production in a given period.

Accounting for the conflicting interest of government and oilcompany by

throwing-in the parameters of production sharing contracts in the decision mix.

The work closest to our proposed research is Skaff (1999) who also considers

various portfolio decisions over time. However our proposed model includes more

features such as probabilistic hydrocarbon price and learning from sequential

drilling, among other things. The sequential drilling is a generalization of Bickel and

Smith (2006) from updating binary (dry/wet) reserve expectations to multiple-level

(e.g., dry, low, medium, high) expectations. This generalization is a useful side

contribution of this paper.

Armed with our multi-faceted DP-based DSS, we perform useful analysis. First,

we demonstrate structural properties of the production decisions, the most important of

which is showing that these decisions are “binary”, in the sense that at a given point in

time either produces the full possible reserve, or produces nothing. This allows reducing

the computer storage requirement and handling large-scale industry-size problems for a

portfolio having a handful of assets. Second, we perform a detailed sensitivity analysis

on the effect of price and reserve uncertainty on the exploration and production

decisions. We observe that high correlation between asset reserves has an important

effect on exploration decisions. In addition, high price volatility shifts decisions to later

in order to benefit from price fluctuations. This is in-line with results from financial

engineering indicating that the value of an option is increasing with its maturity. Here

we have real options and postponing decisions extends maturity.

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The remainder of this paper is organized as follows. Section 2 surveys the related

literature. Section 3 addressed estimation of DSS input parameters. Section 4 presents

the core DSS dynamic programming model. Section 5 presents analysis on the structural

properties of production decisions and computational complexity. Section 6 presents

extensive numerical results and enlightening managerial insights on the effect of price

and reserve uncertainty on the E&P process. Finally, Section 7 concludes the paper and

presents ideas for future research.

2 Literature review

Managing an exploration and production (E&P) portfolio of hydrocarbon

involves many procedures and many decisions that will affect the performance of a

company. The problem is a typical example of companies which have projects with

high return and high risk. The subject was treated by many researchers in different ways

as described below.

Suslick and Schiozer (2004) presented detailed reviews of this literature. They

mention that there are two major techniques used for the evaluation of these projects:

Decision analysis and real options.

Real option evaluation is a technique that utilizes financial options theory and

is usually used for making one time decision (e.g. bid or not, drill or not, lease or not

and to do the valuation of assets). Example of works here includes Lund and Marketing

(1999), Cortazar et al. (2001), and Armstrong et al. (2004). Decision analysis (DA)

utilizes cash flow estimates and optimization theory and typically involves several inter-

related decisions. Our proposed DP model belongs to this category. For this reason, we

focus in the rest of this chapter on reviewing DA models.

Project selection models utilize DA to choose between multiple projects, but

they do not deal with detailed management decisions like exploration and development

and production decisions. Example of works here includes Davidson and Davis (1995),

Motta et al. (2000), Suslick and Furtado (2001), and Keefer (1991).

Mean-Variance models use Markowitz portfolio theory to select the

appropriate combination of E&P projects to reduce the overall variance of the

investments, while meeting a target profit level. Examples of works here include Walls

(2004), and Al-Harthy and Khurana (2008). These papers provide methods to evaluate

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probability distributions by running a simulation that will calculate the variance of the

profit of the project (intra-dependency), and calculate the covariance between projects

(inter-dependency). Even though the models provide novel tools, but they do not help in

business unit field decisions, in terms of scheduling exploration and production

decisions over time.

Dynamic models optimize results by running through all possible states of the

portfolio and valuate the outcomes of various decisions dynamically over time till a

certain boundary state (e.g. decommissioning) that has a predetermined value. Haugen

(1993) uses a different approach then the one used in our model where the E&P

Company is bounded by a contract with downstream company to deliver a certain

amount of gas. Our model is different because it is linked to real world market where

the price is fluctuating and it accounts for learning and production decisions. Aydin

(2008) treats the development of an oil field with a dynamic optimization model. The

model uses Bayes formula to calculate probabilities, and thus will require a large

amount of probability estimation. Most feature in Aydin’s model are captured in our

research with additional downstream considerations of the production decision. Bickel

and Smith (2006) treat the problem of dependency in the outcome of exploration of a

portfolio of reserves. They developed a model that updates probabilities by calculating

joint probabilities for all prospects. These calculations are based on minimizing the

Kullback-Leibler distance relative to a worse-case independent distribution of

outcomes. We adopt a similar Kullback-Leibler based approach for probability in our

model, and we generalize it from binary outcomes (wet/dry) to accommodate multiple

reserve outcomes (e.g. Dry, Medium, and High).

Mixed integer non-linear programs (MINLP) is a method used to find optimal

results of variables relative to a constrained non-linear equation, and variables may be

real or integer. Examples of works here include Frair and Devine (1975), Van den

Heever et al. (2000), Goel and Grossman (2004), Goel et al. (2006). These programs

develop a detailed optimization of the development phase which is inconvenient for an

early stage of the development of an area. During this stage, there is no precise

information about the underground formation, and in consequence, we can’t build

detailed deterministic optimization model of the development phase. In our model, we

reduce the number of decisions in the development phase and we account for the

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possible risk that may arise over the lifetime of the project (e.g. price variability,

dependent random reserve outcomes).

3 Input data estimation

In this section, we will detail the inputs to our model. We show that the

required parameters for our model are measurable and can be estimated with a

reasonable effort.

In Section 3.1, we introduce the contract parameters. In Section 3.2, we

estimate the volatility and growth of gas price. In Section 3.3, we detail the costs

included in the E&P business and explain how they are included in the model. In

Section 3.4, we explain how the reserve probability distribution is currently estimated.

3.1 Contract Parameters

Some of the data inputs that are required for our model can be directly taken

from hydrocarbon contracts or laws. These are typically production sharing agreements

(PSAs) between the government and foreign oil companies. In such agreements, the oil

company bears all the risk of exploration production and shares the revenue of

commercial discoveries with the government. For more details see Younes

(2011).These contract parameters are the royalty ratio, the tax ratio, the exploration

license duration, and the production license duration.

3.2 Estimating the Volatility and Growth of the Gas Price

Modeling the fluctuation of the gas price over time is achieved by collecting

data and estimating the volatility and expected growth rate of gas price. These estimates

are then used to develop a binomial lattice price model.

The Black-Sholes equation is used to calculate the implied volatility of natural

gas price. Using two futures options prices above, we solve for the implied volatility

and the implied current futures price. The Black-Sholes equation is as follows

(Luenberger, 1998).

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⁄ ⁄

√,

√ ,

where

N(x) is the standard normal cumulative density function,

is the current future price, is unknown,

is the strike price, for K = $5and K = $5.5(Two equations),

r is the risk-free rate, r = 0.08% (Source: www.ustreas.gov on June 2010),

T is the maturity time, T = 3 months = 0.25 year,

is the volatility of gas price; is unknown.

In June 2010, the price of futures option with delivery September 2010 and

strike prices K = $5 and K = $5.5 is C = $0.41 and C = $0.248.

The unknowns in the system of two Black-Sholes equations are and . This

gives an implied current futures price of $4.82 and an implied volatility of 50.38% for

Natural Gas.

3.3 Cost Parameter Estimation

In our model, we are assuming that cost parameters are fixed over time. We

were motivated by Goldsmith et al. (2001) in adapting this cost structure. Goldsmith et

al. describe the method used to evaluate the costs of different deep offshore systems

used relative to capital expenditure, operating expenditure, risk expenditure and loss in

production during operation expenditure. Their objective was to decide upon the

optimal system to be installed in a certain field.

Our model has five cost components for each of the assets: exploration cost,

development cost, fixed production cost, unit production cost and abandonment cost,

which are described in the remainder of this section.

The exploration is the expected cost of drilling exploration wells to know if

there is gas in the asset or not. Development cost includes the cost of drilling production

wells, laying network pipes to connect the wells to the gas system, and buying and

installing the subsea system facilities (For deep offshore). Abandonment cost includes

the cost of dismantling the structure and the cost of meeting the environmental

regulation settled by the government. Fixed production cost includes salaries of

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operating staff, maintenance cost and blowoff risk cost. Unit production cost includes

unit treatment cost and the transportation cost.

3.4 Estimating reserve Volume Probability Distribution

The volume of gas that may be discovered in an asset is uncertain. The gas

volume in a compartment can be estimated using the following equation (Hyne 2001).

43560 ∅

Where:

V= Volume of the oil in acre-feet. An acre-foot of volume can hold 43560 scf of

gas

Ф= The porosity of the reservoir (Decimal)

Sg= Gas saturation (gas volume relative to porosity, because gas is also mixed

with water)

R= Technical recovery factor

Bg= Gas formation volume factor (scf/bbl); Bg=1 for a gas reserve

The gas saturation and the technical recovery factors are uncertain before

exploration. By assessing three-point probability distribution for these factors one can

estimate the probability distribution of reserve levels. The reserve probability

distribution is simulated into discrete reserve outcome in the dynamic model.

4 Model and Assumptions

The dynamic model will require probability many inputs to simulate the

uncertainty in the E&P project. As shown in Figure 1, estimates of the volatility of the

Natural Gas price and its expected growth are needed for the price outcome binomial

lattice and they are estimated in Section 3.2. The estimation of contract parameters and

cost parameters is explained in Section 3.1 and Section 3.3; they will be needed for the

dynamic model. Reserve probability distribution estimation is explained in Section 3.4.

In Section 4.1, we will introduce the price outcome binomial lattice sub-model.

In Section 4.2, we introduce Kullback Leibler sub-model, and estimate joint

probabilities out of reserve probability distributions and joint probability distributions.

These two sub-models will feed into the dynamic model to estimate the value of the

project and make good E&P decisions. In Section 4.3, we introduce the dynamic model.

4.1

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46

Figure 2: Binomial lattice model of gas price in the model

Here r' is the rate of increase of different hydrocarbon futures price, which is

different than the risk-free rate because of the storage cost of hydrocarbon (Luenberger

1998, p. 462). r' is calculated from current gas price and gas futures price.

1 .

.1 12.36%

where

: Gas future price for 1 year

: Current gas price.

Table 1: Binomial lattice parameters Natural Gas (r’ = 12.36%, σ = 50%)

u 1.650

d 0.604

qu 0.494

qd 0.506

4.2 The Kullback-Leibler (KL) Probability Update Sub-Model

The exploration of an asset reveals information about the underground

geological layers that will affect the probability distribution of the reserve’s volume of

neighboring assets. Probability distributions of unexplored assets are then updated

relative to the discovered volume in the explored asset.

Figure 3 shows the geological layers in eastern Mediterranean and how the gas

reserves are found on the same geological layers. Actually, during the exploration of

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46

∑ ,, ∈ ∀ ∀ , ∀ (2)

where,

S the set of pair joint probability which are inputs of the sub-model

S the set of pair joint probability which are not inputs of the sub-model

The probability update model utilizes the marginal and pair-wise joint

probability to determine joint probability distribution for all assets in the portfolio. The

all-asset joint probability is used within the DSS model to update reserve distribution

based on the outcome of sequential exploration. The all-asset distribution is determined

via optimization on a worse case basis by minimizing the “distance” to an independent

distribution which ignored correlation between assets, subject to constraints imposed by

the input values of pair-wise probabilities. Specifically, the independent distribution is

given by the following joint mass function

, , … , ∏ , 1, 2, … ,

where is the number of volume outcome of asset i. In addition, the

Kullback-Leibler (K-L) metric is used to measure the distance between the independent

and the all-probability joint distribution, with joint probability mass function ,

which is given by

, ∑ (3)

The all-asset joint distribution is then determined as the solution to the

following mathematical program.

min , (4)

Subject to:

∑ 1 (5)

∑ / (6)

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∑ ⁄ , , , , , ∈ (7)

Solving the above problem directly is demanding mathematically. By

constructing its dual problem which is an exponential objective function formulation

and can be solved as a linear programming, the solution will be easier. In fact, using

Excel solver, we can solve problem having up to 9 assets with 3 reserve outcomes each.

The dual problem is as follows. For each constraint, we assign a dual variable

as shown below.

1 →

/

→

⁄ ,

, , → , ,

Then, the dual problem is as follows,

max

∑ ∗ , ∑ ∑ ⁄

∑ ∑ ∑ ∑ , , , ,, , ∈⁄

(8)

Subject to:

∗ , ,1 ∑ ∑ ⁄

∑ ∑ ∑ ∑ , , , ,, , ∈⁄

(9)

By optimizing the values in vector , we will have ∗ ,

4.3 The Dynamic Model

As shown in Figure 4, the dynamic model will treat E&P decisions. An asset

will start unexplored then exploration decisions will be taken. For a time, the asset will

be explored and after the exploration phase, we will know about the volume of gas in

reserve and we will decide whether to develop it or not. If asset’s development decision

is taken, then for a time, it will be developed. Once asset finishes the development

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phase, it will be a productive asset and production decisions will be taken relative to gas

price level. At the end of the production license or when asset becomes unprofitable,

asset will be decommissioned.

Figure 4: The different upstream phases of the investment of a hydrocarbon asset

The formulation of the dynamic model is composed of an “Exploration” model

and a “Development” sub-model that feeds its results into the exploration model. The

exploration model will go through all possible combinations of exploration decisions.

Once an asset is explored, it will use another model (Development sub-model) to

calculate the profit from developing the asset and producing from it and then

abandoning it. In Section 4.3.1, we introduce the inputs and variables of the dynamic

model. In Section 4.3.2, we introduce and explain the development sub-model DP

formulation. In Section 4.3.3, we introduce and explain the exploration model DP

formulation. In Section 4.3.4, we calculate and show the complexity of the model.

4.3.1 Nomenclature of the Dynamic Programming Formulation

Inputs

: Number of production decisions

, : Current price of the extracted gas

: The volatility of the gas price; used to calculate binomial lattice parameters

: Risk free interest rate

: Royalty Ratio; contract parameter

: Tax Ratio; contract parameter

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: Discount factor on revenues taking into consideration royalty and tax

: Number of assets

: Index of the asset varying between 1 and n

, : Current state of asset i

: Exploration cost of asset i

: Development cost of asset i

: Fix production cost of asset i

: Unit production cost of asset i

: Abandonment cost of asset i

: Deliverability ratio of asset i

: Exploration duration of asset i ( 1)

: Development duration of asset i (z 1)

: Exploration license duration of asset i

: Production license duration of asset i (Includes development time)

: Number of reserve outcomes of asset i

j: Index of the reserve outcome varying between 1 and

: The volume of the reserve outcome j of asset i

: The independent probability of the reserve outcome j of asset i

: Joint probability distribution of all assets in the portfolio

: Multiplicative factor of price going up of the binomial lattice

: Multiplicative factor of price going down of the binomial lattice

: The probability of going up of the binomial lattice

: The probability of going down of the binomial lattice

, : Deliverability of the asset i at time step t

∆ , : Increment of production level of asset i at time t.

Dynamic State Variables

t: Index of the timestep

k: Price level; an integer variable

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: Avectorofs ,

0 ifasset isunexplored1 ifasset isexplored2 ifasset isdeveloped3 ifasset isdecommissionned

∀

: Avectorof , : Discovered reserve index of asset i at timestep t; an integer

variable

,0ifassetreserveisunknown

ifdiscoveredassetreserveindexisthe volume

: Avectorof , ; the remaining timesteps (lag time) till asset i exploration or

development decision is completed at timestep t; an integer variable

: Avectorof , ; The remaining volume in asset i at timestep t; a real valued variable

: Aninformationvectorof , ; it stores the timestep when an asset’s exploration

ended

Decision Variables

: Avectorof , ; Exploration binary decision 0,1 of asset i at timestep t for i=1 to

n; a binary decision.

: Avectorof , ; Development binary decision 0,1 of asset i at timestep t for i=1

to n; a binary decision.

: Avectorof , ; Abandonment binary decision 0,1 of asset i at timestep t for i=1

to n; a binary decision.

: Avectorofx , ; Production decision of asset i at timestep t for i=1 to n; a real

decision.

4.3.2 The Development Sub-Model

The development model will treat the development and production decision

added the fact that the asset may be decommissioned at any time. In the development

model there will be one asset to be studied and not a combination of assets. So, the asset

index is known and it is i and the reserve index is known and it is , . The model starts

at time t and price level k is assigned from the exploration model. Equation 10 is the

objective function. It is the maximization of the profit over the development decision,

abandonment decision and production decision. In the equation, profit is calculated by

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adding revenues from selling gas in this timestep, then subtracting the development

cost, the abandonment cost and the production cost, then adding the maximum expected

future profit over the price of gas using the same formulation.

, , , 0 3,

, , , , I

, , ,

, , , , I

, , , , I

(10)

Where

γ 1 ∝ 1 β (11)

, , (12)

0 01 0 (13)

, , 1 1 2& 00

(14)

0 1 1& 0& 10

(15)

0 3 0 1 0

1 & 2

1& 10

0 1

(16)

0 ∆ 2∆ . . 2& 0& 10

(17)

∆ (18)

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2 13 1 (19)

1 10, 1

(20)

0 , 1

2& 1 1& 1 (21)

Equation 11 calculates the discounting factor on revenues due to royalty and tax.

Equation 12 calculates the gas price at time step t and price level k.

Equation 13 determines whether fixed production cost will be subtracted or not in this.

timestep or not (To be subtracted only if asset is producing).

Equation 14 determines whether development cost will be subtracted or not in this

timestep (To be subtracted only if asset is being developed).

In Equation 15, development decision may be taken if asset is explored and no

abandonment decision is being taken.

In Equation 16, Abandonment decision cannot be taken if asset is decommissioned or it

is being developed or it is producing. Abandonment decision must be taken if either,

production license ends and asset is developed, or there isn’t enough time to develop the

asset before the exploration license ends, or discovered volume in the asset is 0.

Otherwise, abandonment decision may be taken or not.

If asset is productive, the production decisions will take discrete values between 0 and

the deliverability of the asset , relative to the number of production decisions with

a fixed interval between decisions ∆ (See Equation 18).

In Equation 17, production decision may be taken discretely if asset is developed and no

abandonment decision is being taken.

Equation 19 is the state transformation equation of asset state. In this equation, if

development decision is taken, the asset state will be “Developed” or “Being

Developed”, and it will be “Abandoned” if abandonment decision is taken. Otherwise, it

will take current value.

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Equation 20 is the state transformation equation of the remaining time to develop the

asset. The remaining time till the end of development period will take value if

development decision is taken, and it will be of the duration of the development minus

one timestep, otherwise the remaining time will be reduced by one timestep. At any

timestep, the value of this state should not get below 0.

Equation 21 is the state transformation equation of the remaining volume in reserve.

The remaining volume will take the value of the reserve outcome in the next timestep, if

the asset is developed. It will be reduced to zero if the asset is abandoned. Otherwise,

the produced volume will be deducted from the remaining volume.

4.3.3 The Exploration Model

As shown in Equation 22, the objective function of the exploration is to maximize the

profit over exploration decisions of multiple assets. Profit is calculated by subtracting

exploration cost, adding the discounted expected profit of exploring the remaining

unexplored assets in the future by executing the exploration model, and then adding the

expected discounted profit of assets whose exploration has just finished by executing

the development sub-model.

, , , , 0 , 0& , 0∀ ,

,

∈

∑ , , , , ,

∑ ∏ ∈

, , , ,

∑ , , ,

, , , ,

∑ , , , , I ,

∈

(22)

, , , , I ,

, , , , , , , I , , 1& , 1 , 1& 10

(23)

1, 1 ; , 0

Where

is the set of possible combinations of exploration decisions.

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, is the set of indices of assets which exploration ends in the next

timestep

, , , is the set of possible reserve outcome combinations

, , , , ,1 , 1 , 1& , 00

(24)

,0 1 , 00

∀ (25)

,1… , 1& 1 , 1& , 1

,∀ (26)

,1 , 1& 1 , 1& , 1

,∀ (27)

′ ,, , 0& 1 ,

0∀ & ∈ (28)

′ ,

,

, & , 1 , 1& , 0 & 1 ,

0

∀ & ∈ (29)

,

1 , 13 1& , 0

,

∀ (30)

,

1 , 1

0, , 1∀ (31)

, , , is the expected profit of the development of asset i whose discovered

reserve index is w , . Development sub-model will be executed once asset is explored.

Equation 24 determines whether exploration will be deducted in this timestep or not.

As shown in Equation 25, exploration decision can be taken only if asset is unexplored.

46

As shown in Equation 26, reserve index will take value at the end of the

exploration phase. Otherwise, its value remains the same.

Equation 27 shows that if exploration ends in the next timestep, , will take

the value of t+1, otherwise its value will remain the same. , vector is meant to

preserve information about exploration outcome timestep.

During the calculations of probabilities, the volume index vector used for the

calculations of probabilities will be the one that was available at exploration decision

timestep. So, if an outcome of an asset happened after the decision ( z 1 I , ), it

will not be taken into consideration. If multiple outcomes are happening at the same

timestep, probabilities will be calculated independently. Reserve index needed for the

calculation of probabilities will be modified in Equations 28 and 29.

Equation 30 is the state transformation equation of asset state vector. In t+1,

asset i will be either explored or being explored if exploration decision is taken, and it

will be decommissioned if there aren’t enough time to explore the asset in the next

timestep. Otherwise, the state of asset i remains the same.

Equation 31 is the state transformation equation of the remaining exploration

time vector. The remaining time till the end of exploration period will take value if

decision is taken, and it will be the current remaining duration for exploration

subtracted by one, otherwise the remaining time will be subtracted by one. At any state,

this value shouldn’t get below 0.

4.3.4 The Complexity of the model

A state is represented as follows

, , , , ,

The number of states will be calculated for each timestep. The number of price

levels in timestep t is t+1. The number of states of an asset i in a certain timestep t can

be accounted for approximately by counting all possible combinations. These

combinations are divided into different types according to the state of the asset. The

number of states in each type will be calculated,

, : The number of states where the asset is unexplored in timestep t

, : The number of states where the asset is abandoned in timestep t

46

, : The number of states where the asset is being explored in timestep t

, : The number of states where the asset is explored in timestep t

, : The number of states where the asset is being developed in timestep t

, : The number of states where the asset is developed and can produce in timestep t

: Numberofreserveoutcomeslargerthan0inthecalculationofthenumberofstates

As a result, the upper bound number of exploration model states is

1 , , ′ , ,

The upper bound number of development sub-model states is

1 ′′ , , ,

The number of states where asset is unexplored

, 0, , 0, , 0, , 0, , 0

,10

The number of states where asset is being explored

, 1, , 1, , 1… 1, , 0, , 0

,, 1, 0 1

0

The number of states where asset is explored

For exploration model

, 1, , 1… , , 0, , 0, , …

′ ,1, 1 1

0

For development model

, 1, , 1… , , 0, , 0

′′ , 1

0

46

The number of states where asset is being developed

, 2, , 1… , , 1… 1, , 0

,1∗

11

11

0

* It is assumed that each asset has a dry outcome

The number of states where asset is developed and can produce

, 2, , 1… , , 1… , , ⋯ , , …

,1∗

11

1

0

The number of states where asset is decommissioned

, 3, , 0, , 0, , 0, , 0

,1 1 10

Table 2: Number of states of the decoupled model compared with general model

2, 2, 3, 2

Number of

Identical

Assets

Number of

States of

the

Exploration

Model

Number of

States of the

Development

Model

Model

Number

of States

2 4 5 2.80E+02 4.90E+02 7.70E+02

5 4 5 7.67E+04 1.23E+03 7.79E+04

5 4 25 7.67E+04 6.34E+04 1.40E+05

5 8 50 3.39E+07 4.61E+05 3.44E+07

5 10 60 1.95E+08 7.84E+05 1.96E+08

5 15 90 4.17E+09 2.57E+06 4.18E+09

5 16 100 6.73E+09 3.51E+06 6.73E+09

46

5 Structural Properties of the Production Decision in the Development Model

This section represents structural properties of the value function model. In

Section 5.1, we prove that the value function of the development sub-model is convex

in xt. In Section 5.2, we prove that the production decisions are binary, either to produce

at maximum level or not to produce. In Section 5.3, we prove that production decision

is not to be taken when marginal profit is negative. In Section 5.4, we show the

computational enhancements by implementing the theorems in the model. In

Section 5.5, we show the speed of execution of the model on excel (visual basic).

5.1 Value Function of the Development Sub-Model is Convex in xt

Lemma1: , , , , isconvexin

Proof.SeeMelki(2012)1.∎

5.2 Binary Decisions

Theorem 1: Production decisions are binary that is, ∗ ,

Proof.SeeMelki(2012).∎

5.3 Production Decision is not to be Taken When Marginal Profit is Negative

Theorem 2: if at time t and price level k, the marginal profit is negative, ,

∀ , then ∗

Proof.SeeMelki(2012).∎

5.4 Computational Enhancement Results

Theorems 1 and 2 have a great effect on the model. Before, we had a very large

number of decisions and resulting states leading to high storage requirement. These

theorems significantly reduce storage requirement. Table 3 shows the percent decrease

in the number of states relative to taking three discrete timesteps

1 I. G. Melki, “Optimizing Exploration and Production Decisions in Upstream Hydrocarbon Portfolio Management”, American University of Beirut (AUB), 2012.

46

Table 3: The percent decrease in the number of states relative to taking 3 discrete

production decisions

Deliverability\Tp 5 10 15 20 30 0.0005 79.08% 91.76% 93.63% 94.74% 95.80% 0.005 70.41% 83.88% 88.27% 91.23% 93.93% 0.01 62.24% 80.36% 86.34% 90.08% 93.37% 0.05 57.14% 78.55% 85.44% 89.58% 93.15% 0.1 57.14% 78.15% 85.22% 89.47% 93.13%

5.5 The Speed of Execution of the Model

We also tested the duration of the execution of the model on a PC having the

following characteristics.

Processor: Intel (R) Core (TM) i7 CPU Q720 @ 1.60GHz

Installed Memory (RAM): 6.00GB

System type: 64-bit operating system

The duration of the execution of the model is affected by the number of

timesteps and the number of assets. We fixed the duration of the exploration period to

be 5 years and the duration of the development and production period to be 30 years

with timestep duration of 1 year. We increase the number of assets from 1 to 6. The

resulting execution durations are shown in Error! Reference source not found. and

Figure 5.

0

100

200

300

400

500

600

0 2 4 6

Execution Duration (in

seconds)

Number of Assets

46

Figure 5: The execution duration function of the number of assets

As it can be seen in Figure 5, the model runs within few minutes for up to five

assets. For a large number of assets, storage requirement become excessive despite our

efforts to reduce storage in Theorems 1 and 2.

6 Numerical results

In this section, we present numerical results and analyze the behavior of the

exploration, development and production decisions. In section 6.1, we analyze a case of

five assets having a five year exploration period and 30years production period. We

observe exploration results and analyze them. In section 6.2, we observe and analyze the

results of sensitivity on the volatility of the gas price, the interest rate and the

correlation between assets.

6.1 Illustrative Example

A company had just won the bid for developing an offshore block. The contract

with the government licenses the company to explore the block for 5 years and if

hydrocarbon is discovered, the company has the right to develop and produce from asset

for an additional 30years (Inspired from Tamar contract). In return, the government will

get 10% royalty and 40% taxes on hydrocarbon revenues (These values are within

range, see Younes 2011). The company is now looking to exploit the hydrocarbon

reserves in this region.

The company did some geological studies in the area and found out that there

are 5 potential spots where gas reserves can be found, and these assets have correlated

reserve outcomes (Figure 6).

F

6.1.

pro

asse

Tab

acc

Exp

dril

igure 6: Cas

.1 Inputs

Each a

oduction wel

et. The expe

ble 4, and th

cording to ex

ploration), th

lling a produ

se Study Ass

asset will req

ls in order to

ected number

e expected n

xperts that w

he cost of dr

uction well w

sets within th

quire a numb

o produce. T

r of explorat

number of de

we met at LIP

rilling an exp

will be $100M

46

he licensed b

ber of explor

The number o

tion wells fo

evelopment w

PE (Lebanon

ploration we

M. We assum

block and co

ration wells

of these well

or each of the

wells is as sh

n Internation

ll will be $1

me an additi

orrelation bet

and a numbe

ls is relative

e assets is as

hown in Tab

al Petroleum

50M, and th

ional cost of

tween assets

er of

to the size o

s shown in

ble 5. And,

m

he cost of

f $300M will

s

of

l

46

be added to development cost to account for the common facilities to be installed and

for laying pipes to connect the installed subsea system to the existing gas network.

Table 4: The number of exploration wells for each of the assets and the corresponding

exploration cost

Number of

Exploration Wells Exploration Cost

Asset A 1 150M

Asset B 3 450M

Asset C 1 150M

Asset D 1 150M

Asset E 4 600M

Table 5: The number of development wells for each of the assets and the corresponding

development cost

Number of

Development WellsDevelopment Cost

Asset A 4 700M

Asset B 26 2,900M

Asset C 2 500M

Asset D 8 1,100M

Asset E 44 4,600M

In our analysis, we assumed an expected abandonment cost of $25M per drilled

well. Fixed production cost is proportional to the number of production wells. The unit

production cost includes the transportation cost and the treatment cost of gas and it is

assumed to be $0.0005/scf. Inputs to the model are summarized in Table 6, Table 7 and

Table 8.

46

Table 6: Summary of exploration, development, abandonment, fixed production cost

and unit production cost of each of the assets

Asset Name Ce($) Cd($) CA($) Cfp($) Cup($/scf)

Asset A 150M 700M 125M 5M 0.0005

Asset B 450M 2,900M 725M 40M 0.0005

Asset C 150M 500M 75M 5M 0.0005

Asset D 150M 1,100M 225M 10M 0.0005

Asset E 600M 4,600M 1,175M 65M 0.0005

Table 7: General parameters to be used in the estimation of the exploitation of the block

Royalty Ratio, α 10%

Tax Ratio, β 40%

Yearly Risk Free Interest Rate, r 0.30%*

Current Gas Price, p0,0 ($/cfg) 0.004781**

Deliverability 6%***

Volatility of Gas Price, σ 50%****

Gas Price Growth, r’ 12.36%****

u 1.649*****

d 0.607*****

qu 0.496*****

qd 0.504*****

*Treasury bill interest rate for 1year on June 2010 **The price of gas on June 2010 ***Deliverability value used for the development of Tamar gas reserve, Cook

2011 ****Refer to Section 3.2 *****Refer to Section 4.1 for calculation

46

Table 8: Reserve probability distribution of each of the assets

Relative Asset

Name

Discrete Reserve

Outcome Name

Volume in

Reserve (scf)

Marginal

Probability

Expected Reserve

given Wet (scf)

Asset A

A-Low - 0.5 ∗ 0.2 1 ∗ 0.10.2 0.1

0.67T

A-Medium 0.50T p1(2)=0.20

A-High 1.00T p1(3)=0.10

Asset B

B-Low -

4.00T B-Medium 3.00T p2(2)=0.20

B-High 6.00T p2(3)=0.10

Asset C

C-Low -

0.33T C-Medium 0.25T p3(2)=0.20

C-High 0.50T p3(3)=0.10

Asset D

D-Low -

1.33T D-Medium 1.00T p4(2)=0.20

D-High 2.00T p4(3)=0.10

Asset E

E-Low -

6.67T E-Medium 5.00T p5(2)=0.20

E-High 10.00T p5(3)=0.10

The area has not been explored yet, and it has potential gas reserves. Reserve

probability distributions shown in Table 8 are inspired from the reserves already

discovered in Eastern Mediterranean. Compared to Tamar field (8.40T scf of gas) and

Dalit (0.5T scf of gas), these assets are within range.

Assets’ outcomes are either correlated or have a low correlation. Correlated assets have

joint probabilities as shown in

31

Table 9. Assets that have low correlation between each other have joint

probabilities as shown in Table 10.

Table 9: Correlated Assets’ Pair-Joint

Probabilities

Low Medium High

Low

Medium 0.12 0.04

High 0.04 0.03

Table 10: Low Correlation Probabilities

6.1.2 Results

The company will optimize exploration and production decisions using our

DSS model. The DSS model was executed using the inputs in Section 6.1.1 and with

timestep duration of 1year.

The DSS model chose not to explore any asset at t = 0 and t = 1. This behavior

is justified by the fact that gas price is expected to increase (i.e., r’=12.36%) and

postponing decisions will likely increase profit. At t = 2, there are three price level

scenarios to be studied, price increases twice, no price change, and price decreases

twice.

At timestep 2 and for a price level k=2 (i.e., price increased twice), exploration

decisions are as shown in Figure 7. In this case, there is no learning between outcomes

of assets. At this level of the gas price, and knowing the trend of gas price fluctuation, it

is better to wait and get knowledge of the price level. All assets finish their exploration

phase at timestep t=5 when the exploration license ends. As a result, assets B and E are

explored at timestep 3 and assets A, C and D are explored at timestep 4.

Low Medium High

Low

Medium 0.08 0.03

High 0.03 0.02

32

Figure 7: Optimal sequential exploration decisions if project price level is k = 2 at t = 2

If, at timestep t=2, price level is k=0 (no price change), exploration decisions

are as shown in Figure 8.

Figure 8: Optimal sequential exploration decisions if project price level is k = 0 at t = 2

Assets A and C are to be explored first at timestep t=2. At timestep t=3, there

are 2 scenarios whether reserve outcome of asset A is Low or whether it was Medium or

33

High. For a Medium or High reserve outcome of asset A, asset B is explored and the

company benefits from learning of asset A because assets A and B are correlated. For a

Low reserve outcome of asset A, exploration of asset B is not to be happening; knowing

that asset A is dry. At timestep t=3, Asset E is explored regardless of outcomes, it is an

asset that has low correlation with the other assets and it needs two years to be explored.

As for asset D, it will be explored at timestep t=4 and it has low correlation with the

other assets and it needs one year to be explored.

For timestep t=2 and price level k=-2 (decreased twice), exploration decisions’

behavior is the same as of price level k=0 (no price change).

6.2 Sensitivity Analysis on the Model

In this section, we present numerical results analyzing the effect of

environmental parameters especially those involving uncertainty on the profit and

exploration decisions. Uncertainty in this model is represented by the volatility of the

commodity price and the probability distribution of the reserve outcomes. We focus on

analyzing the expected profit and decision making behavior by doing sensitivity

analysis on the volatility of the gas price and the correlation between assets reserve

represented by a parameter which is the value of / , 1

(i.e., the conditional probability of assets j have medium (or high) reserve level given

that asset i has the same reserve level). Accordingly, we calculate the value of joint

probabilities , 1; The other pair joint probabilities are

not inputs to the KL sub-model, they are to be estimated according to the available

information.

We are considering typical general inputs of the model to do the sensitivity

analysis as shown in Table 11.

Table 11: Summary of the base inputs to the sensitivity analysis

Royalty Ratio, α 10%

Tax Ratio, β 40%

Yearly Risk Free Interest Rate, r 0.30%

34

Current Gas Price, p0,0 ($/cfg) 0.004781

Volatility of Gas Price, σ 50%

Gas Price Growth, r’ 12.36%

We are also considering identical assets having the characteristics shown in Table 12.

Table 12: Asset inputs used in the sensitivity analysis

Exploration

Cost

Unit Production

Cost

Exploration

Duration

Exploration License

Duration

150M 0.0005 1 2

We are considering that initial probability distributions are fixed as shown in

Table 13 and we are not doing any sensitivity on them.

Table 13: Reserve outcome inputs used in the sensitivity analysis

Reserve Outcome Name Volume (CFG) Independent Probability

R-L -

R-M 1000B 0.25

R-H 2000B 0.25

Considering a 2 asset independent case ( 0.25), the profit from exploiting

these assets is $2304M.

6.2.1 Sensitivity on Profit

Base inputs used in this analysis represent the current situation of investment in

the E&P of gas reserves. A sensitivity analysis on the base case was done by changing

key model parameters, namely the risk-free rate r, the price growth r’, volatility of gas

price σ, and the correlation parameter ρ. The parameters will be modified discretely

between a minimum value and a maximum value mentioned in Table 14. The vertical

axis of Figure 9 show the percent change in value relative to the base case.

35

Table 14: The range of parameters

Parameter Min. Value Max

Value

r 0% 50.30%

r' 0% 200%

σ 12.50% 250%

ρ 0% 100%

Figure 9: Percent increase/decrease relative to base case profit function the

increase/decrease in the value of each of the parameter (interest rate r, volatility of the

gas price σ, conditional probabilities’ value ρ, gas price growth r’)

‐100%

‐80%

‐60%

‐40%

‐20%

0%

20%

40%

60%

80%

100%

‐100.00% 0.00% 100.00% 200.00% 300.00%

Percent increase/decrease relative to base case

profit

Increase/Decrease in the value of the parameter (r, σ, ρ, r')

r

σ

ρ

r'

36

As shown in Figure 9, in terms of sensitivity magnitude, the expected profit is

highly sensitive to price growth rate r’, followed by the interest rate r, the volatility σ,

and finally the correlation ρ.

6.2.2 Sensitivity on Exploration Decisions for Fixed Gas Price and Fixed Interest

Rate

In this section, the price of gas is fixed ( 0 ’ 0%). For independent

assets, exploration decisions are as shown in Figure 10.

Figure 10: Optimal exploration decisions for a project with two identical independent

assets (r’=0%, 0, r=0.3%)

Both assets are explored at t=0 because of the time value of money. In this

case, postponing exploration and production decisions will generate less profit.

If the two assets’ outcomes are perfectly correlated (ρ=1), then exploration

decisions will be as shown in Figure 11. So, the first asset is explored first, and the

second asset is explored in the next timestep when the first asset’s exploration ends with

a medium or high reserve outcome (Note that if the first asset’s exploration outcome is

dry, the second asset will not be explored).

37

Figure 11: Optimal exploration decisions for a project with two identical assets (ρ=1,

r’=0%, ~0, r=0.3%)

As shown in Figure 12, a one way sensitivity analysis is done. At a certain

positive correlation, learning will start profiting the project. Unlike what is presented in

Figure 9, profit in this case increased by 35% due to the increase in the value of .

Figure 12: Percentage increase in profit relative to two independent asset profit function

of ρ (r’=0%, ~0, r=0.3%)

6.2.3 Sensitivity on Exploration Decisions for Volatile Gas Price and Interest Rate

In this section, we are analyzing the effect of having two uncertainties in the

model. We analyze how decisions are taken relative to the volatility of gas price and the

correlation between reserve outcomes of different assets.

0%

5%

10%

15%

20%

25%

30%

35%

0.2 0.4 0.6 0.8 1

Percentage

increase in

profit

relative to two independent

assets

ρ

38

For σ = 50%, the exploration of both assets ended when their exploration

license ended. So, exploration decisions were “shifted to the end” as shown in Figure

13, in order to benefit from the potentially high price levels in the production phase.

Figure 13: Optimal exploration decisions for a project with two identical

Figure 14 analyzes the trade-off between volatility σ and learning effects on the

exploration decisions timing prospective. Exploration decisions for both assets are being

taken at the beginning for low volatility and low correlation. As the value of ρ increases,

exploration decisions will become subsequent and learning will occur. For high

volatility, exploration decisions will be shifted to the end of exploration license. As

aforementioned, this occurs in order to benefit from a potential prices increase under

high volatility.

Figure 14: Optimal exploration decision behavior of two assets in two way sensitivity

on σ and ρ (r’=0%, r=0.3%, 2 assets)

By increasing the number of similar assets to three, similar results will be

reflected, but learning effect becomes more important as shown in Figure 15. Learning

σ\ρ 0.25 0.3 0.40 0.50 0.60 0.70 0.80 0.90 0.99

0.10%

2.00%

5.00%

10.00%

15.00%

20.00%

30.00%

50.00%

80.00%

100.00%

140.00%

200.00%

All Explorations at t=0

Learning

Exploration Decisions at t=1

39

happens (as reflected by sequential exploration decisions) over a wider range of

volatility σ, and correlation ρ.

Figure 15: Optimal exploration decision in two way sensitivity on σ and ρ

Next, we investigate the effect of the risk-free rate. It is interesting to study the

effect of an increase in the interest rate (0.3%), as it may be the case in practice.

6.2.4 Sensitivity on Exploration Decisions for a Volatile Gas Price and Different

Values of Interest Rate

Considering the base case with fixed price, we increased the value of the interest rate to

5%. On the other hand, and as shown in Figure 16, exploration decisions behaved

almost the same as in Figure 14 where interest rate is 0.3%.

Figure 16: Optimal exploration decisions of two assets in two way sensitivity on σ and ρ

σ\ρ 25% 30% 40% 50% 60% 70% 80% 90% 99%

0.10%

2.00%

5.00%

10.00%

15.00%

20.00%

30.00%

50.00%

80.00%

100.00%

140.00%

200.00%

Learning



σ\ρ 0.25 0.3 0.40 0.50 0.60 0.70 0.80 0.90 0.99

0.10%

2.00%

5.00%

10.00%

15.00%

20.00%

30.00%

50.00%

80.00%

100.00%

140.00%

200.00%

Learning



40


on σ and ρ (r’ = 0%, r = 15%, 2 assets)

At high interest rates, exploration costs have a higher weight in the present

value of the expected profit. Therefore, exploration/production decisions are shifted to

the future to benefit from high potential prices that could offset the high present value of

the exploration cost. As shown in Figure 17, exploration decisions are taken at timestep

t = 1 for lower value of volatility than that shown in Figure 14 (i.e., interest rate is

0.3%). As the interest rate increases to 25%, as shown in Figure 18, projects with low

volatility and low correlation are becoming unprofitable, and for higher volatility the

model is choosing to wait for a probable increase in price to take exploration decisions.

Exploration decisions’ behavior is shown in Figure 19.


on σ and ρ (r’ = 0%, r = 25%, 2 assets)

σ\ρ 0.25 0.3 0.40 0.50 0.60 0.70 0.80 0.90 0.99

0.10%

2.00%

5.00%

10.00%

15.00%

20.00%

30.00%

50.00%

80.00%

100.00%

140.00%

200.00%


Exploration

Decisions at t=0 Learning

σ\ρ 0.25 0.3 0.40 0.50 0.60 0.70 0.80 0.90 0.99

0.10%

2.00%

5.00%

10.00%

15.00%

20.00%

30.00%

50.00%

80.00%

100.00%

140.00%

200.00%

No Exploration Decisions, Assets

are unprofitable


Learning makes the whole project

profitable

41

Figure 19: Optimal exploration decisions for a project with two identical independent

assets (r’ = 0%, 10%, r = 25%)

6.2.5 Decision Making for High Price Growth Rate

In this subsection, we observe the results of sensitivity analysis where the

expected price growth is high (r’ = 12.36%). In this case, for a two asset portfolio, when

the interest rate is 0.3%, all exploration decisions are at t=1 (i.e., the second timestep) as

shown in Figure 20. When interest rate increases to 25% (Figure 21), then for low

volatility values and high correlation between asset, learning is happening; otherwise,

exploration decisions are taken at t = 1.

Figure 20: Optimal exploration decision behavior of assets in two way sensitivity on σ

and ρ (r’=12.36%, r=0.3%, 2 assets)

σ\ρ 0.25 0.30 0.40 0.50 0.60 0.70 0.80 0.90 0.99

12.00%

15.00%

20.00%

30.00%

50.00%

80.00%

100.00%

140.00%

200.00%


42


and ρ (r’=12.36%, r=25%, 2 assets)


and ρ (r’=12.36%, r=0.3%, 3 assets)


and ρ (r’=12.36%, r=25%, 3 assets)

Learning becomes more important when the number assets increases, this is

demonstrated between Figure 20 and Figure 22, and between Figure 21 and Figure 23.

σ\ρ 0.25 0.30 0.40 0.50 0.60 0.70 0.80 0.90 0.99

12.00%

15.00%

20.00%

30.00%

50.00%

80.00%

100.00%

140.00%

200.00%


Learning

σ\ρ 25% 30% 40% 50% 60% 70% 80% 90% 99%

12.00%

15.00%

20.00%

30.00%

50.00%

80.00%

100.00%

140.00%

200.00%

LearningExploration Decisions at t=1

σ\ρ 25% 30% 40% 50% 60% 70% 80% 90% 99%

12.00%

15.00%

20.00%

30.00%

50.00%

80.00%

100.00%

140.00%

200.00%

Learning

Exploration Decision at t=1

43

7 Conclusions

In this research, we develop a dynamic model that handles multiple decisions

over time. The model is capable of simulating hydrocarbon price fluctuation and

learning from exploration drilling. We experimented on the model and observed that

profit in E&P projects are sensitive to the expected price growth and volatility of the gas

price. On the other hand, E&P projects are negatively sensitive to the interest rate. We

observed while doing sensitivity analysis on the model that exploration decision taking

is affected the expected gas price growth, the volatility of the gas price, the interest rate

and the correlation between assets. For a fixed price (or low expected price growth and

low volatility) and high interest rate, the decisions (i.e., exploration and production

decisions) will be shifted to earlier timesteps. As for having a high expected price

growth or high volatility of the price, decisions will be shifted to later timesteps. Even

with high interest, it is more profitable to wait for later timesteps to benefit from a

probable price growth. Finally, for a high correlation between assets, exploration

decisions will be successive to benefit from learning.

References

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a real-option approach for portfolio management of upstream hydrocarbon...

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