# Portfolio Real Option Based on Trigeminal Tree Model in Sustainable Utilization of Exploration Asset

Authors: Yiping Wu; Jianjun Wang; Qing Wang; Qian Li; Haowu Li; Ningning Zhang; Qingchao Cao
Journal: Wireless Communications and Mobile Computing
 (2022)

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## Abstract

Uncertainty of crude oil market causes the value of oil and gas exploration assets to fluctuate upward, remain unchanged, and fluctuate downward. So exploration project investment is usually chartered by multistages. This brings many challenges to investment decisions regarding the sustainable utilization of exploration assets. To avoid the investment risk of exploration projects, a portfolio option method based on the trigeminal tree model was proposed. By analysing the real options involved in oil and gas exploration assets, the investment process is divided into European options and American options. The pricing model of portfolio options is constructed by the trigeminal tree method, and sequential compound diagram and compound option diagram are built. Volatility is predicted based on the GARCH model, and then, the trinomial tree of the underlying asset value, evolution chart, sequential compound option chart, and portfolio option chart is established, respectively. Finally, the project option value is calculated. The application shows that the ROV of oil and gas exploration project of company A is 1.39 million US$ (USMM$), and the project value is 279.87 USMM$. Compared with the binary tree model, the model is superior to the traditional option pricing method in the application effect, which provides a scientific basis for investment decision-making.


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## Body

## 1. Introduction

Real options are derived from financial options; therefore, their classification can refer to real financial options. According to the different execution times of option contracts, real options are divided into American and European options. After Stewart Myers introduced the real option theory into the investment field, the option theory and method have made great progress in solving the problem of option pricing and project risk [1–3] and have been extended to the fields of land development, mine operation, and mining investment [4–10]. For example, Acciaro applied the real option model to calculate the correlation between LNG investment cost and fuel price and suggested that decision-makers should adopt advanced technology to keep LNG price at a certain level [11]. However, most of the current research results continue the premise and assumption of financial option pricing method; it is unreasonable and undesirable. Some scholars have begun to focus on more complex compound option pricing model, but the application of compound option pricing in oil investment has not been reported publicly.Due to the uncertainty of oil and gas market, oil and gas exploration project investment is usually chartered by multistages. In order to reduce uncertain market risks, decision-makers must formulate flexible investment strategies to deal with complex and changeable markets [12]. According to the stages of oil and gas projects and the mutual influence of each stage, the compound options are involved in the decision-making of oil and gas assets and can be divided into two types: sequential options and compound options. The oil and gas exploration projects witness the common situations: when the oil market presents optimistic, the enterprise will expand oil and gas production and carry out the expansion period to obtain more profits; if the oil market turns down, the decision-maker can sell some shares to reduce the loss as much as possible. At this time, there are contraction options [13–18]. If the market continues to depress, the decision-maker can sell the assets and exercise the abandonment option. There are three options at three points in time; that is to say, at same time, the decision-maker has the right to choose a variety of options [19], which constitutes a compound option. The paper innovates means to establish the pricing model of portfolio options by the trigeminal tree method.


## 2. Theory and Methodology

At present, the research on trigeminal tree option pricing model is generally limited to single option, but there are three forms of oil and gas asset value in global crude oil market: upward fluctuation, constant fluctuation, and downward fluctuation, so investment decision-making is more complex. To solve this problem, the trinomial tree model extends from single option to compound option, which includes time series option and compound option.


### 2.1. Single Option Model in Investment Decision

St represents the underlying asset value of time t change. In order to construct the trigeminal tree model, the continuous variable St needs to be discretized into equal intervals of N time intervals Δt=T/N. The change of the asset value of the internal standard in a certain time interval is shown in the following Figures 1 and 2, and the option value changes are as follows in the chart.Figure 1
The change of underlying assets.Figure 2
The change of option values.According to the definition of mathematical expectation,(1)ES=PuSu+PmS+PdSd,(2)ES2=PuSu2+PmS2+PdSd2,(3)ES3=PuSu3+PmS3+PdSd3.According to the hypothesis, the change of the value of the underlying asset follows the geometric Brownian motion, and the expected rate of return of the underlying assetu is the risk-free rate of return r. Therefore,
(4)ES=SerΔt.According to Ito’s lemma, the differential equations ofS2 and S3 are obtained.
(5)dS2S2=2r+σ2dt+2σdz,(6)dS3S3=3r+3σ2dt+3σdz.Formulas (2) and (3) are discretized, and the following results are obtained:
(7)ΔS2S2=2r+σ2dt+2σΔz,(8)ΔS3S3=3r+σ3dt+3σΔz.The integral solution can be obtained:(9)ES2=S2e2r+σ2Δt,(10)ES3=S3e3r+3σ2Δt.Formulas (1)–(4), (9), and (10) in parallel can be immediately derived the following equations:
(11)Pu+Pm+Pd=1,(12)ud=1,(13)erΔt=uPu+Pm+dPd,(14)e2r+σ2Δt=u2Pu+Pm+d2Pd,(15)e3r+3σ2Δt=u3Pu+Pm+d3Pd.The equations are solved and simplified by Taylor’s expansion formula, and the calculation formulas of rising factoru, falling factor d, and risk neutral probabilities Pu, Pm, and Pd are obtained as follows:
(16)u=eσ3Δt=1+σ3Δt+3σ2Δt2+oΔt,(17)d=e−σ3Δt=1−σ3Δt+3σ2Δt2+oΔt,(18)Pu=16+r−σ22Δt12σ2+oΔt,(19)Pm=23+oΔt,(20)Pd=16−r−σ22Δt12σ2+oΔt.


### 2.2. Sequential Compound Option Model in Investment Decision

Sequential option is a kind of real option due to the stage of investment. It is assumed that an investment project can be divided into two stages. The first stage is the initial investment stage with a period of 1 year and the cost requires C1; the second stage is the expansion investment stage, with a period of 2 years and the cost is C2. It is known that the logarithmic current value of the project’s net income is zero, and there is no risk in the future. According to the above parameters, use formulas (16)–(20); the ascending factor u, the descending factor D, and the risk neutral probabilities Pu, Pm, and Pd can be derived from formulas (1)~(5). This is the basis of the sequential option pricing model.First of all, establish the value of the underlying assets trigeminal tree (Figure3).Figure 3
The trinomial tree of multistage compound of underlying asset.Then, calculate value of the second stage, so as to establish the net asset value chart of long-term options (Figure4).The determination of nodes in the net asset value chart needs to use the back steeping technology, which is obtained from the end in turn.Figure 4
The trinomial tree of net value of underlying asset.The value of the terminal node is determined and tookFS0u3,3 as an example.
(21)Fs0u3=maxu3s0−c2,0.Refer to this rule to determine the value of the last column node. The intermediate node is selected by comparing the value of exercise option and maintenance option and tookFS0u3,2 as an example:
(22)FS0u2,2=maxu2S0−C2,e−rΔtPu⋅FS0u3,3+Pm⋅FS0u2,3+Pd⋅F,where C2 is the option execution cost, e−rΔtPu⋅FS0u3,3+Pm⋅FS0u2,3+Pd⋅FS0u,3 is the value when the option continues to be hold in the node, u2S0−C2 is the return on the exercise of the option, and the larger one is selected to determine the remaining node values.Finally, on the basis of trigeminal tree with the long-term option net asset value, the option valuation chart is established, and the total option value is composed of the above option and the shorter option in the first stage. Then, the trigeminal tree of sequential option valuation is obtained to get the trigeminal tree of compound option valuation (Figure5).The analysis of the trigeminal tree of sequential option valuation depends on the one of net asset value of long-term option.VS0u,1 is taken as an example to determine the value of terminal nodes:
(23)VS0u,1=maxFS0u,1−C1,0,Figure 5
Trinomial tree of sequential options valuation.According to this rule, the end node value is determined.The option value of the initial node is derived from the later node.(24)VS0,1=e−rΔtPu⋅VS0u,1.If there is an intermediate node, the rule can be still used: max (the value of holding the option, the income from exercising the option) to determine the option valuation at all times, and finally obtain the option value at the initial time.Finally, establish the joint option valuation chart of sequential conforming to the option (Figure6), and the value of the sequential option is VS0,0.Figure 6
The trinomial tree of sequential compound options.


### 2.3. Portfolio Option Model

Portfolio option in oil and gas investment decision-making is a kind of compound option due to the multiple choices of investment schemes. If the oil company adopts strategic option strategy to reduce risks, in any time in the next three years, the decision-makers can choose to reduce it to a fraction of the previous business scale according to the market, so as to save costa. Or expand the business scale several times at the cost of B. Or give up the current project and harvest the residual value C. The present value of the expected income of the oil company is S0, the implied volatility of logarithm yield of expected return cash flow is σ, and the nonrisk interest rate of the next five years is r. It can be seen from the case that the company holds a portfolio option including expansion option, curtailment option, and abandonment option in the next three years and still uses the trigeminal tree method to construct the pricing model of portfolio option.Firstly, according to the known parameters, Taylor’s expansion formulas (1)–(5) are applied to calculate the rising factor, decline factor, and risk neutral probability, and the compound option pricing model could be established. The expansion factor b is known, as well the expansion cost B, the contraction factors a, the reduced balance A, and the abandoned residual value C. Based on the above data, a trigeminal tree of the underlying asset value is established (Figure 7).Figure 7
The trinomial tree of underlying asset.Secondly, establish the trigeminal tree of portfolio option valuation.Decision rules of the end node are given and takenFS0u2,3 as an example.
(25)FS0u2,3=maxb⋅S0u2−B,a⋅S0u2s,where b⋅S0u2−B represents the expansion value and a⋅S0u2+A,C,S0 are the reduced value. C is the value of exercising abandonment options, S0u2 is the value of maintaining the status quo, and the maximum can be selected as the end node value.Decision rules of intermediate node value are given and takenFS0u,1 as an example.
(26)FS0u,1=maxb⋅S0u−B,a⋅S0u+A,C,e−rΔtPu⋅FS0u2,2+Pm⋅FS0u,2+Pd⋅FS0,2,where e−rΔtPu⋅FS0u2,2+Pm⋅FS0u,2+Pd⋅FS0,2 represents the value of the point to maintain the status quo. The value of each node is determined in turn by the back stepping technique, and finally, trigeminal tree valuation chart of the portfolio option is obtained (Figure 8).Figure 8
The trinomial tree of combined compound options.Finally, the price of the portfolio option isFS0,0−S0.Brent oil is one of the most widely traded crude on the futures market, accounting for about 65 percent of global crude volume. The expansion option described in the paper is a European option with a term of two years. Therefore, the data period is set to two years, namely, September 30, 2018, to 2020.10.1, which meets the research needs.


### 2.4. Option Volatility Estimation

The reason why the trigeminal tree model shows a trend of up and down changes is that the underlying asset value has volatility. The parameter of two volatilityσ, which represents the change of asset value [20], is the key factor affecting the value of options. In the option pricing model, the accuracy of σ is a key to evaluate the value of investment projects accurately. In the study of real options, four kinds of volatility are mainly involved: implied volatility, actual volatility, historical volatility, and forecast volatility. Forecast volatility is commonly used in the study of option value calculation. Generally speaking, forecasting volatility is different from historical volatility; it comes from the understanding of historical volatility and people’s prediction and judgment of the actual situation.In the actual trading process, the historical volatility is often used as the basic data for forecasting, and then based on the continuous correction of quantitative data, the forecast volatility is finally determined. In this paper, EViews software is applied to establish GARCH (1, 1) model to predict volatility, the formula is shown in (27).
(27)σ2=α0+αεt−12+βσt−12,where εt−12 is the lag term of residual square and σt−12 is the lag term of conditional variance. The parameters α and β in the model are nonnegative, and α0 is a positive number.


## 2.1. Single Option Model in Investment Decision

St represents the underlying asset value of time t change. In order to construct the trigeminal tree model, the continuous variable St needs to be discretized into equal intervals of N time intervals Δt=T/N. The change of the asset value of the internal standard in a certain time interval is shown in the following Figures 1 and 2, and the option value changes are as follows in the chart.Figure 1
The change of underlying assets.Figure 2
The change of option values.According to the definition of mathematical expectation,(1)ES=PuSu+PmS+PdSd,(2)ES2=PuSu2+PmS2+PdSd2,(3)ES3=PuSu3+PmS3+PdSd3.According to the hypothesis, the change of the value of the underlying asset follows the geometric Brownian motion, and the expected rate of return of the underlying assetu is the risk-free rate of return r. Therefore,
(4)ES=SerΔt.According to Ito’s lemma, the differential equations ofS2 and S3 are obtained.
(5)dS2S2=2r+σ2dt+2σdz,(6)dS3S3=3r+3σ2dt+3σdz.Formulas (2) and (3) are discretized, and the following results are obtained:
(7)ΔS2S2=2r+σ2dt+2σΔz,(8)ΔS3S3=3r+σ3dt+3σΔz.The integral solution can be obtained:(9)ES2=S2e2r+σ2Δt,(10)ES3=S3e3r+3σ2Δt.Formulas (1)–(4), (9), and (10) in parallel can be immediately derived the following equations:
(11)Pu+Pm+Pd=1,(12)ud=1,(13)erΔt=uPu+Pm+dPd,(14)e2r+σ2Δt=u2Pu+Pm+d2Pd,(15)e3r+3σ2Δt=u3Pu+Pm+d3Pd.The equations are solved and simplified by Taylor’s expansion formula, and the calculation formulas of rising factoru, falling factor d, and risk neutral probabilities Pu, Pm, and Pd are obtained as follows:
(16)u=eσ3Δt=1+σ3Δt+3σ2Δt2+oΔt,(17)d=e−σ3Δt=1−σ3Δt+3σ2Δt2+oΔt,(18)Pu=16+r−σ22Δt12σ2+oΔt,(19)Pm=23+oΔt,(20)Pd=16−r−σ22Δt12σ2+oΔt.


## 2.2. Sequential Compound Option Model in Investment Decision

Sequential option is a kind of real option due to the stage of investment. It is assumed that an investment project can be divided into two stages. The first stage is the initial investment stage with a period of 1 year and the cost requires C1; the second stage is the expansion investment stage, with a period of 2 years and the cost is C2. It is known that the logarithmic current value of the project’s net income is zero, and there is no risk in the future. According to the above parameters, use formulas (16)–(20); the ascending factor u, the descending factor D, and the risk neutral probabilities Pu, Pm, and Pd can be derived from formulas (1)~(5). This is the basis of the sequential option pricing model.First of all, establish the value of the underlying assets trigeminal tree (Figure3).Figure 3
The trinomial tree of multistage compound of underlying asset.Then, calculate value of the second stage, so as to establish the net asset value chart of long-term options (Figure4).The determination of nodes in the net asset value chart needs to use the back steeping technology, which is obtained from the end in turn.Figure 4
The trinomial tree of net value of underlying asset.The value of the terminal node is determined and tookFS0u3,3 as an example.
(21)Fs0u3=maxu3s0−c2,0.Refer to this rule to determine the value of the last column node. The intermediate node is selected by comparing the value of exercise option and maintenance option and tookFS0u3,2 as an example:
(22)FS0u2,2=maxu2S0−C2,e−rΔtPu⋅FS0u3,3+Pm⋅FS0u2,3+Pd⋅F,where C2 is the option execution cost, e−rΔtPu⋅FS0u3,3+Pm⋅FS0u2,3+Pd⋅FS0u,3 is the value when the option continues to be hold in the node, u2S0−C2 is the return on the exercise of the option, and the larger one is selected to determine the remaining node values.Finally, on the basis of trigeminal tree with the long-term option net asset value, the option valuation chart is established, and the total option value is composed of the above option and the shorter option in the first stage. Then, the trigeminal tree of sequential option valuation is obtained to get the trigeminal tree of compound option valuation (Figure5).The analysis of the trigeminal tree of sequential option valuation depends on the one of net asset value of long-term option.VS0u,1 is taken as an example to determine the value of terminal nodes:
(23)VS0u,1=maxFS0u,1−C1,0,Figure 5
Trinomial tree of sequential options valuation.According to this rule, the end node value is determined.The option value of the initial node is derived from the later node.(24)VS0,1=e−rΔtPu⋅VS0u,1.If there is an intermediate node, the rule can be still used: max (the value of holding the option, the income from exercising the option) to determine the option valuation at all times, and finally obtain the option value at the initial time.Finally, establish the joint option valuation chart of sequential conforming to the option (Figure6), and the value of the sequential option is VS0,0.Figure 6
The trinomial tree of sequential compound options.


## 2.3. Portfolio Option Model

Portfolio option in oil and gas investment decision-making is a kind of compound option due to the multiple choices of investment schemes. If the oil company adopts strategic option strategy to reduce risks, in any time in the next three years, the decision-makers can choose to reduce it to a fraction of the previous business scale according to the market, so as to save costa. Or expand the business scale several times at the cost of B. Or give up the current project and harvest the residual value C. The present value of the expected income of the oil company is S0, the implied volatility of logarithm yield of expected return cash flow is σ, and the nonrisk interest rate of the next five years is r. It can be seen from the case that the company holds a portfolio option including expansion option, curtailment option, and abandonment option in the next three years and still uses the trigeminal tree method to construct the pricing model of portfolio option.Firstly, according to the known parameters, Taylor’s expansion formulas (1)–(5) are applied to calculate the rising factor, decline factor, and risk neutral probability, and the compound option pricing model could be established. The expansion factor b is known, as well the expansion cost B, the contraction factors a, the reduced balance A, and the abandoned residual value C. Based on the above data, a trigeminal tree of the underlying asset value is established (Figure 7).Figure 7
The trinomial tree of underlying asset.Secondly, establish the trigeminal tree of portfolio option valuation.Decision rules of the end node are given and takenFS0u2,3 as an example.
(25)FS0u2,3=maxb⋅S0u2−B,a⋅S0u2s,where b⋅S0u2−B represents the expansion value and a⋅S0u2+A,C,S0 are the reduced value. C is the value of exercising abandonment options, S0u2 is the value of maintaining the status quo, and the maximum can be selected as the end node value.Decision rules of intermediate node value are given and takenFS0u,1 as an example.
(26)FS0u,1=maxb⋅S0u−B,a⋅S0u+A,C,e−rΔtPu⋅FS0u2,2+Pm⋅FS0u,2+Pd⋅FS0,2,where e−rΔtPu⋅FS0u2,2+Pm⋅FS0u,2+Pd⋅FS0,2 represents the value of the point to maintain the status quo. The value of each node is determined in turn by the back stepping technique, and finally, trigeminal tree valuation chart of the portfolio option is obtained (Figure 8).Figure 8
The trinomial tree of combined compound options.Finally, the price of the portfolio option isFS0,0−S0.Brent oil is one of the most widely traded crude on the futures market, accounting for about 65 percent of global crude volume. The expansion option described in the paper is a European option with a term of two years. Therefore, the data period is set to two years, namely, September 30, 2018, to 2020.10.1, which meets the research needs.


## 2.4. Option Volatility Estimation

The reason why the trigeminal tree model shows a trend of up and down changes is that the underlying asset value has volatility. The parameter of two volatilityσ, which represents the change of asset value [20], is the key factor affecting the value of options. In the option pricing model, the accuracy of σ is a key to evaluate the value of investment projects accurately. In the study of real options, four kinds of volatility are mainly involved: implied volatility, actual volatility, historical volatility, and forecast volatility. Forecast volatility is commonly used in the study of option value calculation. Generally speaking, forecasting volatility is different from historical volatility; it comes from the understanding of historical volatility and people’s prediction and judgment of the actual situation.In the actual trading process, the historical volatility is often used as the basic data for forecasting, and then based on the continuous correction of quantitative data, the forecast volatility is finally determined. In this paper, EViews software is applied to establish GARCH (1, 1) model to predict volatility, the formula is shown in (27).
(27)σ2=α0+αεt−12+βσt−12,where εt−12 is the lag term of residual square and σt−12 is the lag term of conditional variance. The parameters α and β in the model are nonnegative, and α0 is a positive number.


## 3. Case Study

### 3.1. Case Background

At present, the oil company A has an oil and gas exploration block abroad, covering an area of 360 km2. Three oil fields have been discovered with 230 million tons of oil reserves. In 2019, the crude oil production reached 3.5 million tons, and there are 5 prospective traps. Undiscovered oil resources are estimated to be 90 million tons. At the beginning of 2020, due to the outbreak of 2019-nCoV, the price of Brent crude oil dropped to below 40 US$/bbl. The profit of company A fell sharply, and it lost 240 USMM$ in 2020. In order to ensure the sustainable development, decision-makers plan to increase the crude oil production to 5.5 million tons in 2022 when the oil price rises to 60 $/bbl. The total exploration wells, development wells, and surface construction related to the increase of production will cost 55 USMM$. If the operation benefit is not optimistic after the production increase, the company will choose to farm out 50% of the block shares in the next three years of production rising to obtain 140 million US dollars of income, or all exploration blocks to obtain 280 million US dollars. Assuming that the production period of the block is 10 years, the remaining value D of the block is $6 million. According to the characteristics of the oil and gas industry, the nonrisk interest rate in the next five years is determined as 4.5%.


### 3.2. Option Analysis

The overall investment plan of the above case contains both sequential options and compound options, and the whole investment process can be divided into two stages: the first stage is two-year expansion option, which can be regarded as European rising option; the second stage is the compound option with three valid years, which composed of contraction option and abandonment option and can be regarded as American option. The execution of the first stage expansion option is the basis of the second stage compound option. Therefore, the two-stage expansion option and the portfolio option constitute the time series portfolio option.Company A forecasts the cash flow and income in the next 10 years according to its own operation and management situation and the expectation of future income (Table1).Table 1
Net cash flow of company A in the next ten years.
Year20202021202220232024202520262027202820292030NCF/USMM$-24034.473.4229.4104.643.5-4.634.3115156.673.4According to the net cash flow (NCF) method, company A discounts the income of the next 10 years to the initial node. NPV will be as follows:(28)NCF=∑i=110ai1+Rt,where R is the discount (10%) and D is the residual value of the block.In NCF method, NCF represents the whole value of exploration investment project, but in option pricing method, NCF is only the value at the starting point in trigeminal tree model of underlying assets, and the final value of exploration investment project is the sum of NCF and option value of project.


### 3.3. Estimation of Volatility

Only the impact of global oil prices on expected revenue is considered in the paper, and other factors are out of consideration in the process of production and operation currently. Using GARCH (1BI) and considering the influence of historical data, the volatility of the freight rate within the validity period of the option is predicted as the value ofσ in the option pricing.


### 3.4. Date Processing

Brent oil price is selected as the sample data, and the oil price index seriesL is established in EViews. The premise of using GARCH model to predict volatility is that the sample data is stationary, the volatility of crude oil market is large, and the sequence L is obviously not stationary.Therefore, the logarithmic difference of oil price is processed, that is,(29)rt=1nLt−1.On this basis, the logarithmic rate of returnr of the oil price index is constructed, and the time series of the logarithmic rate of return is obtained (Figure 9).Figure 9
The time series of logarithmic rate of return.It can be observed that under the influence of the new crown epidemic, the fluctuation of logarithmic yield of oil price is mainly concentrated in the first half of 2020, especially in May 2020, and its fluctuation is quite large, and the fluctuation in the rest of the time is relatively stable.By using the numerical analysis function of EViews, the frequency analysis and statistics of the sample data are carried out, and the column statistical chart of logarithmic rate of return is obtained (Figure10).Figure 10
The time series of logarithmic rate of return.It can be seen from Figure10 that the Jarque-Bera test value of sequence r is 1011.031, the corresponding p value is 0, the skewness is -1.442367, and the peak value is 17.99977. The sample data shows an obvious trend of left thick tail, so the rejection sequence is subject to the original hypothesis of normal distribution.


### 3.5. Stationarity Test

In order to investigate the stationarity, the unit root test of the sequencer is carried out (Figure 11).Figure 11
Adf unit root test.As can be seen from Figure11, the statistical value of t is -73.91646 and the corresponding p value is close to 0, indicating that the sequence r is stable.


### 3.6. Autocorrelation and Partial Autocorrelation Test

In order to judge the degree of correlation between sample data and analyse the time series characteristics of data fluctuations, the Correlogram module is used to test the autocorrelation and partial autocorrelation of oil price return series (Figure12).Figure 12
The Correlogram result of seriesr.As can be seen, the autocorrelation coefficient and partial autocorrelation coefficient of the sequence are close to 0, and the correspondingp values are greater than 0.05. Therefore, it accepts the original hypothesis that r has no significant correlation at 5% significance level.


### 3.7. ARCH Effection Test

The heteroscedasticity is tested by the square correlation diagram of the residual [21], the logarithmic return series of oil price is dehomogenized, the residual sequence is obtained, a new residual sequence r is established, and the Correlogram module is applied to verify the autocorrelation of the residual series (Figure 13).Figure 13
The Correlogram result of residual square series.According to the observation ofp value, there is autocorrelation in the square sequence of residual error, so there is ARCH effection, and the GARCH model can be established.


### 3.8. Establishment of GARCH (1, 1) Model

Using the equation function of EViews, the calculation results of GARCH (1, 1) model are obtained (Figure14).Figure 14
The result of GARCH (1, 1) regression.There are three parameters ofα0 and αand β estimated by GARCH (1, 1), which have passed t-test at the significance level of 5%. According to formula (30), α0=0.007935, α=0.539368, β=0.309374, and the available volatility is forecast as follows:
(30)σ2=a0+aεt2+βσt−12=0.007935+0.539368εt2.


### 3.9. Volatility Calculation

On the basis of the influence of historical data, the volatility of selling oil price during the period of option validity is predicted as the value ofσ in option pricing. The model is mainly bulit by the econometric software EViews. The two curves in Figure 15 represent the confidence level of the rate of return R, which is 95% (Figure 15). In the interval, with the increase of forecast periods, the confidence zone widens slightly. The lower curve shows the prediction results of conditional variance (Figure 16 and Table 2). As can be seen, the predicted value slowly increases and finally converges to 0.052, so the volatility is 0.2280.Figure 15
Historical data chart of variance.Figure 16
Forecasting chart of variance.Table 2
Forecasting sheet of variance.
ForecastActualForecast sampleAdjusted sampleRoot mean squared errorMean absolute errorMean abs. percent errorTheil inequality coefficientBias proportionVariance proportionCovariance proportionStartEndStartEndRFR1/01/201712/30/20181/08/201712/30/20180.1461320.071640108.71680.9470790.008455NANAAccording to formulas (1)–(6), the parameters for building the model are shown in Table 3.Table 3
Parameters of compound options.
ParametersS0TσμdPuPmPdabABCValue278.4752.8%1.4840.6740.1910.6670.1420.51.214055280UnitUSMM$YearUSMM$USMM$USMM$


### 3.10. Calculation of Option Value

#### 3.10.1. Modelling the Trinomial Tree of the Underlying Asset Value

In this case, the effective period of the total compound option is 5 years. Therefore, there are 6 nodes in 0-5. According to the rising factorμ and the falling factor D, the trinomial tree of the underlying asset value is established (Figure 17).Figure 17
The trinomial tree of underlying asset value.


#### 3.10.2. Modelling the Evolution Chart of Underlying Assets

Due to the existence of the expansion period in the second year, the value of the underlying asset increases by 1.2 times of the previous one. Therefore, the value of the underlying asset of node 2 is changed to 1.2 times of the corresponding node in Figure4, and the subsequent nodes are pushed forward in turn to build the evolution chart of the underlying asset (Figure 18).Figure 18
Evolution chart of underlying assets.


#### 3.10.3. Modelling the Trigeminal Tree of Portfolio Option Valuation

The compound option consists of expansion one in the first stage and portfolio one in the second stage. Firstly, the trigeminal tree of option valuation is built based on the second stage of portfolio options. In the last column, a decision rule is introduced: max (the balance of exercise contraction options -1/2 of the underlying asset value of the corresponding node, the balance of the exercise and abandonment option—the underlying asset value of the corresponding node, 0), so as to determine the end node value of the trigeminal tree of portfolio option valuation. Then, the decision rule is introduced in nodes 2, 3, and 4: max (balance of exercise contraction options -1/2 of the underlying asset value of the corresponding node, the balance of the exercise and abandonment option—the value of the underlying asset of the corresponding node and the value of the option to continue to be held), where the value of the continued holding option is the discount value of the expected option return at the current time node according to the nonrisk interest rate.TakingFS0u4,4 as an example, the decision rules of this point are as follows:
(31)FS0u4,4=maxA−12u4S0,C−u4S0,e−rΔtPu⋅FS0u5,5+Pm⋅FS0u4,5+Pm⋅FS0u4,5+Pd⋅FS0u3,5,where e−rΔtPu⋅FS0u5,5+Pm⋅FS0u4,5+Pd⋅FS0u3,5 is the discount value of the future income of the options held at note FS0u4,4, so as to determine the node value and establish the valuation chart of portfolio options (Figure 19).Figure 19
Trigeminal tree of portfolio option valuation.


#### 3.10.4. Modelling Trigeminal Tree Valuation Chart of Sequential Options

Company A holds expansion options in the first stage, so it is necessary to construct the valuation chart of overall conforming options based on the trigeminal tree valuation chart of portfolio options and the impact of expansion options. For nodes 0, 1, and 2, the decision rule max (the value of the exercise expansion option, 0) is used at the end node to determine the valuation of the last column. Since the expansion options of the company belong to European options, the value of the intermediate node does not need to be compared, and the value of the maintenance option can be calculated directly. Given calculation formula (11) of intermediate node value, option value is determined by VS0u,1.(32)VS0u,1=e−rΔtPu⋅VS0u2,2+.Finally, the option value of the initial node, that is, the value of the overall compound optionVS0,0, is derived from backward technique of trigeminal tree (Figure 20).Figure 20
Trigeminal tree valuation chart of sequential options.Through trigeminal tree pricing model, the value of the compound option ROV is 1.39 USMM$, and the value of oil and gas exploration project is NCF + ROV =279.87 USMM$.


## 3.1. Case Background

At present, the oil company A has an oil and gas exploration block abroad, covering an area of 360 km2. Three oil fields have been discovered with 230 million tons of oil reserves. In 2019, the crude oil production reached 3.5 million tons, and there are 5 prospective traps. Undiscovered oil resources are estimated to be 90 million tons. At the beginning of 2020, due to the outbreak of 2019-nCoV, the price of Brent crude oil dropped to below 40 US$/bbl. The profit of company A fell sharply, and it lost 240 USMM$ in 2020. In order to ensure the sustainable development, decision-makers plan to increase the crude oil production to 5.5 million tons in 2022 when the oil price rises to 60 $/bbl. The total exploration wells, development wells, and surface construction related to the increase of production will cost 55 USMM$. If the operation benefit is not optimistic after the production increase, the company will choose to farm out 50% of the block shares in the next three years of production rising to obtain 140 million US dollars of income, or all exploration blocks to obtain 280 million US dollars. Assuming that the production period of the block is 10 years, the remaining value D of the block is $6 million. According to the characteristics of the oil and gas industry, the nonrisk interest rate in the next five years is determined as 4.5%.


## 3.2. Option Analysis

The overall investment plan of the above case contains both sequential options and compound options, and the whole investment process can be divided into two stages: the first stage is two-year expansion option, which can be regarded as European rising option; the second stage is the compound option with three valid years, which composed of contraction option and abandonment option and can be regarded as American option. The execution of the first stage expansion option is the basis of the second stage compound option. Therefore, the two-stage expansion option and the portfolio option constitute the time series portfolio option.Company A forecasts the cash flow and income in the next 10 years according to its own operation and management situation and the expectation of future income (Table1).Table 1
Net cash flow of company A in the next ten years.
Year20202021202220232024202520262027202820292030NCF/USMM$-24034.473.4229.4104.643.5-4.634.3115156.673.4According to the net cash flow (NCF) method, company A discounts the income of the next 10 years to the initial node. NPV will be as follows:(28)NCF=∑i=110ai1+Rt,where R is the discount (10%) and D is the residual value of the block.In NCF method, NCF represents the whole value of exploration investment project, but in option pricing method, NCF is only the value at the starting point in trigeminal tree model of underlying assets, and the final value of exploration investment project is the sum of NCF and option value of project.


## 3.3. Estimation of Volatility

Only the impact of global oil prices on expected revenue is considered in the paper, and other factors are out of consideration in the process of production and operation currently. Using GARCH (1BI) and considering the influence of historical data, the volatility of the freight rate within the validity period of the option is predicted as the value ofσ in the option pricing.


## 3.4. Date Processing

Brent oil price is selected as the sample data, and the oil price index seriesL is established in EViews. The premise of using GARCH model to predict volatility is that the sample data is stationary, the volatility of crude oil market is large, and the sequence L is obviously not stationary.Therefore, the logarithmic difference of oil price is processed, that is,(29)rt=1nLt−1.On this basis, the logarithmic rate of returnr of the oil price index is constructed, and the time series of the logarithmic rate of return is obtained (Figure 9).Figure 9
The time series of logarithmic rate of return.It can be observed that under the influence of the new crown epidemic, the fluctuation of logarithmic yield of oil price is mainly concentrated in the first half of 2020, especially in May 2020, and its fluctuation is quite large, and the fluctuation in the rest of the time is relatively stable.By using the numerical analysis function of EViews, the frequency analysis and statistics of the sample data are carried out, and the column statistical chart of logarithmic rate of return is obtained (Figure10).Figure 10
The time series of logarithmic rate of return.It can be seen from Figure10 that the Jarque-Bera test value of sequence r is 1011.031, the corresponding p value is 0, the skewness is -1.442367, and the peak value is 17.99977. The sample data shows an obvious trend of left thick tail, so the rejection sequence is subject to the original hypothesis of normal distribution.


## 3.5. Stationarity Test

In order to investigate the stationarity, the unit root test of the sequencer is carried out (Figure 11).Figure 11
Adf unit root test.As can be seen from Figure11, the statistical value of t is -73.91646 and the corresponding p value is close to 0, indicating that the sequence r is stable.


## 3.6. Autocorrelation and Partial Autocorrelation Test

In order to judge the degree of correlation between sample data and analyse the time series characteristics of data fluctuations, the Correlogram module is used to test the autocorrelation and partial autocorrelation of oil price return series (Figure12).Figure 12
The Correlogram result of seriesr.As can be seen, the autocorrelation coefficient and partial autocorrelation coefficient of the sequence are close to 0, and the correspondingp values are greater than 0.05. Therefore, it accepts the original hypothesis that r has no significant correlation at 5% significance level.


## 3.7. ARCH Effection Test

The heteroscedasticity is tested by the square correlation diagram of the residual [21], the logarithmic return series of oil price is dehomogenized, the residual sequence is obtained, a new residual sequence r is established, and the Correlogram module is applied to verify the autocorrelation of the residual series (Figure 13).Figure 13
The Correlogram result of residual square series.According to the observation ofp value, there is autocorrelation in the square sequence of residual error, so there is ARCH effection, and the GARCH model can be established.


## 3.8. Establishment of GARCH (1, 1) Model

Using the equation function of EViews, the calculation results of GARCH (1, 1) model are obtained (Figure14).Figure 14
The result of GARCH (1, 1) regression.There are three parameters ofα0 and αand β estimated by GARCH (1, 1), which have passed t-test at the significance level of 5%. According to formula (30), α0=0.007935, α=0.539368, β=0.309374, and the available volatility is forecast as follows:
(30)σ2=a0+aεt2+βσt−12=0.007935+0.539368εt2.


## 3.9. Volatility Calculation

On the basis of the influence of historical data, the volatility of selling oil price during the period of option validity is predicted as the value ofσ in option pricing. The model is mainly bulit by the econometric software EViews. The two curves in Figure 15 represent the confidence level of the rate of return R, which is 95% (Figure 15). In the interval, with the increase of forecast periods, the confidence zone widens slightly. The lower curve shows the prediction results of conditional variance (Figure 16 and Table 2). As can be seen, the predicted value slowly increases and finally converges to 0.052, so the volatility is 0.2280.Figure 15
Historical data chart of variance.Figure 16
Forecasting chart of variance.Table 2
Forecasting sheet of variance.
ForecastActualForecast sampleAdjusted sampleRoot mean squared errorMean absolute errorMean abs. percent errorTheil inequality coefficientBias proportionVariance proportionCovariance proportionStartEndStartEndRFR1/01/201712/30/20181/08/201712/30/20180.1461320.071640108.71680.9470790.008455NANAAccording to formulas (1)–(6), the parameters for building the model are shown in Table 3.Table 3
Parameters of compound options.
ParametersS0TσμdPuPmPdabABCValue278.4752.8%1.4840.6740.1910.6670.1420.51.214055280UnitUSMM$YearUSMM$USMM$USMM$


## 3.10. Calculation of Option Value

### 3.10.1. Modelling the Trinomial Tree of the Underlying Asset Value

In this case, the effective period of the total compound option is 5 years. Therefore, there are 6 nodes in 0-5. According to the rising factorμ and the falling factor D, the trinomial tree of the underlying asset value is established (Figure 17).Figure 17
The trinomial tree of underlying asset value.


### 3.10.2. Modelling the Evolution Chart of Underlying Assets

Due to the existence of the expansion period in the second year, the value of the underlying asset increases by 1.2 times of the previous one. Therefore, the value of the underlying asset of node 2 is changed to 1.2 times of the corresponding node in Figure4, and the subsequent nodes are pushed forward in turn to build the evolution chart of the underlying asset (Figure 18).Figure 18
Evolution chart of underlying assets.


### 3.10.3. Modelling the Trigeminal Tree of Portfolio Option Valuation

The compound option consists of expansion one in the first stage and portfolio one in the second stage. Firstly, the trigeminal tree of option valuation is built based on the second stage of portfolio options. In the last column, a decision rule is introduced: max (the balance of exercise contraction options -1/2 of the underlying asset value of the corresponding node, the balance of the exercise and abandonment option—the underlying asset value of the corresponding node, 0), so as to determine the end node value of the trigeminal tree of portfolio option valuation. Then, the decision rule is introduced in nodes 2, 3, and 4: max (balance of exercise contraction options -1/2 of the underlying asset value of the corresponding node, the balance of the exercise and abandonment option—the value of the underlying asset of the corresponding node and the value of the option to continue to be held), where the value of the continued holding option is the discount value of the expected option return at the current time node according to the nonrisk interest rate.TakingFS0u4,4 as an example, the decision rules of this point are as follows:
(31)FS0u4,4=maxA−12u4S0,C−u4S0,e−rΔtPu⋅FS0u5,5+Pm⋅FS0u4,5+Pm⋅FS0u4,5+Pd⋅FS0u3,5,where e−rΔtPu⋅FS0u5,5+Pm⋅FS0u4,5+Pd⋅FS0u3,5 is the discount value of the future income of the options held at note FS0u4,4, so as to determine the node value and establish the valuation chart of portfolio options (Figure 19).Figure 19
Trigeminal tree of portfolio option valuation.


### 3.10.4. Modelling Trigeminal Tree Valuation Chart of Sequential Options

Company A holds expansion options in the first stage, so it is necessary to construct the valuation chart of overall conforming options based on the trigeminal tree valuation chart of portfolio options and the impact of expansion options. For nodes 0, 1, and 2, the decision rule max (the value of the exercise expansion option, 0) is used at the end node to determine the valuation of the last column. Since the expansion options of the company belong to European options, the value of the intermediate node does not need to be compared, and the value of the maintenance option can be calculated directly. Given calculation formula (11) of intermediate node value, option value is determined by VS0u,1.(32)VS0u,1=e−rΔtPu⋅VS0u2,2+.Finally, the option value of the initial node, that is, the value of the overall compound optionVS0,0, is derived from backward technique of trigeminal tree (Figure 20).Figure 20
Trigeminal tree valuation chart of sequential options.Through trigeminal tree pricing model, the value of the compound option ROV is 1.39 USMM$, and the value of oil and gas exploration project is NCF + ROV =279.87 USMM$.


## 3.10.1. Modelling the Trinomial Tree of the Underlying Asset Value

In this case, the effective period of the total compound option is 5 years. Therefore, there are 6 nodes in 0-5. According to the rising factorμ and the falling factor D, the trinomial tree of the underlying asset value is established (Figure 17).Figure 17
The trinomial tree of underlying asset value.


## 3.10.2. Modelling the Evolution Chart of Underlying Assets

Due to the existence of the expansion period in the second year, the value of the underlying asset increases by 1.2 times of the previous one. Therefore, the value of the underlying asset of node 2 is changed to 1.2 times of the corresponding node in Figure4, and the subsequent nodes are pushed forward in turn to build the evolution chart of the underlying asset (Figure 18).Figure 18
Evolution chart of underlying assets.


## 3.10.3. Modelling the Trigeminal Tree of Portfolio Option Valuation

The compound option consists of expansion one in the first stage and portfolio one in the second stage. Firstly, the trigeminal tree of option valuation is built based on the second stage of portfolio options. In the last column, a decision rule is introduced: max (the balance of exercise contraction options -1/2 of the underlying asset value of the corresponding node, the balance of the exercise and abandonment option—the underlying asset value of the corresponding node, 0), so as to determine the end node value of the trigeminal tree of portfolio option valuation. Then, the decision rule is introduced in nodes 2, 3, and 4: max (balance of exercise contraction options -1/2 of the underlying asset value of the corresponding node, the balance of the exercise and abandonment option—the value of the underlying asset of the corresponding node and the value of the option to continue to be held), where the value of the continued holding option is the discount value of the expected option return at the current time node according to the nonrisk interest rate.TakingFS0u4,4 as an example, the decision rules of this point are as follows:
(31)FS0u4,4=maxA−12u4S0,C−u4S0,e−rΔtPu⋅FS0u5,5+Pm⋅FS0u4,5+Pm⋅FS0u4,5+Pd⋅FS0u3,5,where e−rΔtPu⋅FS0u5,5+Pm⋅FS0u4,5+Pd⋅FS0u3,5 is the discount value of the future income of the options held at note FS0u4,4, so as to determine the node value and establish the valuation chart of portfolio options (Figure 19).Figure 19
Trigeminal tree of portfolio option valuation.


## 3.10.4. Modelling Trigeminal Tree Valuation Chart of Sequential Options

Company A holds expansion options in the first stage, so it is necessary to construct the valuation chart of overall conforming options based on the trigeminal tree valuation chart of portfolio options and the impact of expansion options. For nodes 0, 1, and 2, the decision rule max (the value of the exercise expansion option, 0) is used at the end node to determine the valuation of the last column. Since the expansion options of the company belong to European options, the value of the intermediate node does not need to be compared, and the value of the maintenance option can be calculated directly. Given calculation formula (11) of intermediate node value, option value is determined by VS0u,1.(32)VS0u,1=e−rΔtPu⋅VS0u2,2+.Finally, the option value of the initial node, that is, the value of the overall compound optionVS0,0, is derived from backward technique of trigeminal tree (Figure 20).Figure 20
Trigeminal tree valuation chart of sequential options.Through trigeminal tree pricing model, the value of the compound option ROV is 1.39 USMM$, and the value of oil and gas exploration project is NCF + ROV =279.87 USMM$.


## 4. Comparison and Discussion

In order to further clarify the advantages and disadvantages of trigeminal tree model, according to the parameters in Table3, the binary tree method is applied to recalculate the assets, and the results are compared and analysed.The first step is to construct the underlying asset grid and use table (Table4) instead of binary tree. According to the given parameters, the underlying asset grid is obtained by backward technique of trigeminal tree.Table 4
Underlying asset sheet of compound options.
Grid series012345Asset value/USMM$278.47352.54446.32565.04715.34905.62—219.96278.47352.54446.32565.04——173.75219.96278.47352.54———137.24173.75219.96————108.41137.24—————85.63The second step is to construct the evolution grid of the underlying assets, as shown in Table5.Table 5
Evolution grid of the underlying asset.
Grid series012345Asset value/USMM$278.47352.54535.58678.05858.411086.75—219.96334.17423.06535.59678.05——208.50263.96334.17423.06———164.69208.50263.96————130.09164.70—————102.76The third step is to construct the compound option value grid, as shown in Table6.Table 6
Compound option value sheet.
Grid series012345Asset value/USMM$2.7400000—6.2003.1300——14.04335.227.080———115.3171.5016.04————149.91115.30—————177.24The compound option valueV of the asset is 2.74 USMM$, that is, the Expected Net Cash Flow ENCF is as follows: ENCF=NCF+OV=278.47+2.74=281.21USMM$.However, the asset value obtained by the trigeminal tree pricing model is 279.87 USMM$, which is lower than that of the binary tree model. This is because the value of the underlying asset remains unchanged by the binomial tree model, which is a kind of waiting option and does not change over time, so the option value is overestimated. This change is fully considered in the trigeminal tree model, and the value of the asset falls with the continuous forward discount. Therefore, the accuracy of the trigeminal tree model is higher than that of the binary tree model.However, the trigeminal tree option pricing model is based on the known accurate future income and investment cost. In the process of estimating the option value, it is difficult to predict the exact value of income and cost. Because of the volatility of investment outlook, the range of the rise and fall in the underlying asset value itself has certain uncertainty. Therefore, although the trigeminal tree model considers random value variance, it ignores the uncertainty of the underlying asset itself.In order to effectively define the above uncertainties, it is necessary to introduce the fuzzy theory into the real option investment decision-making model of oil and gas exploration assets in the future research and establish a trigeminal tree model based on fuzzy theory and real options, so as to improve the shortcomings of the classic trigeminal tree model.


## 5. Conclusions

(1)
The paper constructs the trigeminal tree option pricing model. Compared with the binary tree model, it has faster convergence speed and higher calculation accuracy because of allowing more possibilities within the time interval. By analysing the possibility that the underlying assets remain unchanged, it further clarifies the existence of investment extension option, which has better applicability in calculating the value of oil and gas exploration assets(2)
The GARCH (1, 1) model is built to predict the volatility based on the historical data to improve the accuracy of the parameters(3)
Due to many factors such as resources and oil prices, the value of frontier exploration assets and low to medium exploration assets is highly uncertain. The trigeminal tree model proposed in this paper can be widely used in the value evaluation of the above assets, effectively improve forecasting precision, and provide scientific basis for investment decision-making



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*Source: 4662460-2022-04-27.xml*