What is this paper about?

Use option valuation theory to develop a new approach to valuing leases for

offshore petroleum

Theoretical and practical problems not present in applying options to financial

assets

Why is valuation important?

• Firms perform valuations as inputs to their bidding process

• Government uses to establish presale reservation prices and to study effect of policy changes on revenue (underestimates)

• Bidding process involves billions of dollars

important to obtain accurate valuations

3 Stages

• Exploration– seismic and drilling activity

• quantities of hydrocarbon reserves

– costs of bringing them out

• Development– put equipment in place to extract oil

• platforms, production wells

– converts undeveloped reserves to developed

• Extraction– Use the installed capacity to take the hydrocarbons

out of the ground

Resultsfavorable?

Relinquish?

Exploration

Development

Relinquish?

Extraction

YES

YES

YES

NO

NO

NO

OPTION #1

OPTION #2

DCF Approach

• Specify distributions for– Exploration costs, quantities of hydrocarbon reserves,

development costs, hydrocarbon prices, and operating costs

• An analyst determines whether it is optimal for the firm to explore, develop and extract

• Analyst makes assumptions about timing, and rate of extraction

• The time path of cash flows determined• Involves multivariate Monte Carlo simulations

Major Weaknesses of DCF

• The proper timing is not transparent

• Different assessments of future statistical distributions by different companies

• Choosing correct set of risk-adjusted discount rates is a difficult task

• Very complex and costly

• The assessments of geological and cost distributions can wary widely

Tract Valuation by the Option Valuation Approach

Characteristics of the Stages Exploration Development Extraction

Valuation Petroleum Reserve Market Equilibrium Valuing Undeveloped Reserves Valuing Unexplored Tracts Exploration and Development Lags Optimal Investment Timing Comparative Statics

Comparison of Option Valuation and Discounted Cash Flow Approaches

Tract Valuation by the Option Valuation Approach

Characteristics of the Stages-Exploration

The exploration stage consists of the option to make the exploration expenditures and to receive undeveloped reserves. It’s very similar to a stock option.

The main difference is the uncertainties ( the quantities of hydrocarbons ) in the exploration stage.

Tract Valuation by the Option Valuation Approach

Characteristics of the Stages-Exploration

We can represent the exploration stage as the option to spend the exploration cost , and receive the expected value of undeveloped reserves

where =random quantity of recoverable hydrocarbons in the tract =per unit development cost, a function of quantity =current value of a unit of developed hydrocarbon reserves =probability distribution over the quantity of hydrocarbons =current per unit value of undeveloped reserves given the current per unit value of a developed reserve and per unit development cost =current date =expiration date

€

X *(V ) = QX(V ,T − t;D(Q))dF(Q)∫€

E

€

Q

€

D(Q)

€

V

€

F(Q)

€

X(V ,T − t;D(Q))

€

t

€

T

Tract Valuation by the Option Valuation Approach

Characteristics of the Stages-Development

Once exploration has provided an indication of the quantity of hydrocarbons, the leaseholder has the option to pay the development costs and install the productive capacity.

Characteristics of the Stages-Extraction

The leaseholder has the option to extract the hydrocarbons after he has exercised the development option.

Tract Valuation by the Option Valuation Approach

Valuation-Petroleum Reserve Market Equilibrium

In equilibrium, the expected net payoff from holding a developed reserve must compensate the owner for opportunity cost of investing in that reserve.

Assume the rate of return to owner follows the diffusion process

where

=the number of units of petroleum in a developed reserve

=the value of a unit of developed reserve

=the instantaneous per unit time net payoff from holding the reserve

=the required rate of return to the owner

=the instantaneous per unit time standard deviation of the rate of return

=an increment to diffusion process

€

Rtdt /BtVt =α υ* dt +σ υ dzυ

€

αυ*

€

συ

€

dzυ

€

Bt

€

Vt

€

Rt

Tract Valuation by the Option Valuation Approach

Valuation-Petroleum Reserve Market Equilibrium

comes from two sources

(1) The profits from production

(2) The capital gain on holding the remaining petroleum

Assume a developed reserve follow an exponential decline

Then the net payoff can be written as

where the net payoff is over a short interval . is the after-tax

Operation profit from selling a unit of petroleum

€

Rt

€

dt

€

Pt

€

dBt = −γBtdt

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Rtdt = γBtPtdt{ } + (1− γdt)Bt (Vt + dVt ) − BtVt{ }

Tract Valuation by the Option Valuation Approach

Valuation-Petroleum Reserve Market Equilibrium

The process for the value of a producing developed reserve

=the payout rate of the producing developed reserve

=the expected rate of capital gain€

dV

V= (α υ

* −δ)dt +σ υ dzυ

=α υ dt +σ υ dzυ

where

α υ =α υ* −δt

δt = γ Pt −Vt[ ] /Vt

€

δt

€

αυ

Tract Valuation by the Option Valuation Approach

Valuation-Petroleum Reserve Market Equilibrium

Comparison of Variables Pricing Models of Stock Call Options and Undeveloped Petroleum Reserves

Valuing Undeveloped Reserves: What is it & Why do we need it?

• Given a tract that has been explored, we find X(V, T-t, D)

• Firms need to value reserves to make decisions

• It is done before valuing an unexplored tract

Comparison of the valuation with Stock Call Options

• Current stock price

• Variance of rate of return

• Exercise price

• Time to expiration

• Riskless rate of interest

• Dividend

• Value of developed reserve discounted for developed lag

• Variance of rate of change of the value of a developed reserve

• Per unit development cost

• Relinquishment requirement

• Riskless rate of interest

• Net production revenue less depletion

Finding X(V, T-t, D)

• Invoke standard arbitrage arguments by replicating the undeveloped reserve’s payoff by holding a portfolio of developed reserves and riskless bonds.

– Holding nonproducing developed reserves - feasible but inefficient

– Holding producing developed reserves - works

Problem

• We use the Black-Scholes price as the price of the call option– The price of the contingent claim should equal the

cost of a strategy that replicates the returns of that claim

• But the option would earn a subnormal rate of return – not an equilibrium situation

• Excess of writers to buyers – drives down call price

The second one works

• The holder of a producing developed reserve earns a fair rate of return

• The payout is identical to a proportional dividend on a stock

• The PDE for valuing the option on stock can be used for valuing an undeveloped reserve

Invoking standard arbitrage arguments

• It is difficult to effect the actual arbitrage • We use an equilibrium analysis given by

Constantinides [1978]

• The equilibrium model of the petroleum reserves in brought in through δ

Boundary conditions

• X(Vt, T-t, D) = Vt – D if Ct = Ct* and Cs<Cs* for all s<t– Ct = Vt / D – Ct* Boundary that maximizes solution– Ct hits Ct* from below for the first time

• Ct* can be used for any lease since it is independent of V and D

Ct* - a closer look

• Hitting boundary decreases with time– Option value

decreases with time– No time no option

value– Vt - D is not positive

anymore

Boundary Conditions

• X(0,T-t,D) = 0 for all t– If there is no value for the developed hydrocarbon

reserve then there is no value for the undeveloped reserve

• X(VT, 0, D) = max[0, VT – D] if Cs < Cs* for every s < T

• There are no closed forms for the solution to the PDE and Ct* – Use numerical solutions

Valuing Unexplored Tracts

• Complications due to the properties of the development option and optimal development timing

– Assume that development begins immediately after successful exploration – collapse the two options

– More later…

Finding W(V, T-t, S)

• From the development option, we have

• Recall,

• In an exploration option, you pay and get

• Or paying and getting

• Value of unexplored tract is

The collapsing technique

• With no geological uncertainty, S > D if V/S exceeds hitting boundary then so will V/D

• With geological uncertainty this is not the case

• We get a lower bound to the true option value

Exploration and Development lags

• Let t be the length of the lag• The value of the claim at t to receive a developed reserve

at t+t is

• By beginning development at t, the firm gets this claim • The underlying asset in both these options is the claim• Also, Vt follows a diffusion process• We replace Vt with Vt

^

^

Optimal Investment Timing

• Begin development or exploration the first time that Ct hits Ct

* from below

• Insights:– Reserves with low investment costs will hit the

boundary before those with high investment costs – Herfindahl’s equilibrium

– Properties with shorter investment lags will be explored or developed before those with longer lags

Comparison of OV and DCF approaches

• Reduces the amount of information required

– Estimation of future developed reserve values– Determination of risk-adjusted discount rates– Explicit modeling of the extraction stage

Data Sources For Results

• Calculate the market value for offshore petroleum tracts awarded to industry in federal lease sale no. 62 in November 18, 1980

– 21 of the 38 tracks compared (available data)– Data on the tracts they used is protected by privacy

laws– Paper only looks at bonus bidding with a fixed 16

2/3% royalty on the tracks• Used for tracts valued at <= $10,812,077• Company owes 16 2/3% in amount or value of production

saved, removed or sold

Who Benefits?

– Relatively low royalty system used since the OCS Lands Act in 1953

• Negative- results in greater risks to the lessee from finding a dry hole

• Positive- more rewards (lower contingency payments to the government) if a commercial field is discovered.

– If a dry hole is found, then the tract is unusable and the company loses money

– If a commercial field is discovered• the company reaps the benefits for the first year• the next year the government recategorizes the tract

Calculating royalty on large tracts

eg. sliding scale royalty– higher royalty rates for larger reservoirs with

higher production rates

Rj= b[ln(Vj/s)] (in Millions)

Rj is percent royalty due in quarter jb= 13.0

Vj is the value of production in quarter j

USGS

• Information obtained by USGS for tracks– Mean and variance for quantities of recoverable

• oil reserves• condensate reserves• gas reserves

– Probability that the tract is dry– Expected

• exploration cost• development cost

– USGS estimate of tract value (estimated using DCF calculation with the above as input parameters)

Inputs into Valuation Equation: Developed reserve value

• Compare with current market value– $12/barrel of oil– 1/6 cost of a barrel of oil for an mcf of gas

• $2/mcf as benchmark• $3/mcf from private bakers for latter 1980s

– Unavailable information to authors would be available to firms: break down the valuation based on the quality of the tract based on market value

• Hydrocarbon quality• Cost structure • Tax regime

Inputs into Valuation Equation: Variance

• Variance of the rate of change in the value of developed reserves

• Techniques– Estimate based on past data on market values of

developed reserves• Neg: market value data is not publicly available regularly

enough to estimate the variance directly• Estimate based on Gruy et al. [1982]: developed reserve

prices tend to be 1/3 of crude oil prices• Therefore, use the variance of the rate of change of crude oil

prices as a proxy for the variance of the rate of change of developed reserve prices

Inputs into Valuation Equation: Variance (cont.)

• Representative period: 1974-1980– Periods of crisis– Periods of tranquility

• Using monthly data from 1974-1980: σ2=0.02019 -> σ=0.142 To account for increase in perceived uncertainty: (Jacoby and Paddock[1983]) σ2=0.0625 -> σ=0.250

• Per Barrel Crude oil Wellhead price ranges implicit in standard deviations: Year 0= 1980 @ $36/barrel

• 95% confidence

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Inputs into Valuation Equation: Expected Stages 1&2 Costs

• Expected exploration costs before tax (USGS)• 10% of the costs are depreciated (not taxable)

Dj=Aj[6Qoj+Qgj]

Qoj = recoverable oil reserves on jth tract

Qgj = recoverable gas reserves on jth tractAj= tract-specific scaling parameter that considers water depth and drilling depth = 2/3 (Mansvelt Beck and Wiig [1977])

[ ] represent total reserve volume measured in terms of cubic feet of gas equivalent (BTU conversion factor: 1 barrel = 6 mcf)

Inputs into Valuation Equation: Expected Stages 1&2 Costs

Calculating the track-specific parameters Aj using a fitting procedure.

Step 1: Take second-order Taylor Expansion of Dj

= variances of oil quantities= variances of gas quantities= covariance between the above two

Note: bars represent expected values (which can’t keep)

Step 2: Arbitrary assumption: σogi= 0.5 σoj σoj

Solve for Aj to get:

Use the track specific means for distributions Dj, Qoj, Qgj Result: Track-specific development cost functions to approximate the

true developtment cost functions (information is protected by USGS)

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D j = A j 6Qoj +Qgj( )β

+ 18σ oj2 +σ gj

2

2+ 6σ ogj

2 ⎛

⎝ ⎜

⎞

⎠ ⎟A jβ (β −1) 6Qoj +Qgj( )

β −2

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A j =D j

6Qoj +Qgj( )β

+ 18σ oj2 +σ gj

2

2+ 3σ ojσ gj

⎛

⎝ ⎜

⎞

⎠ ⎟β (β −1) 6Qoj +Qgj( )

β −2

Option Valuation Comparisons

• Comparison with USGS Estimates– Differences should be due primarily to differences

in the financial valuation techniques

– To increase the fairness of the comparisons, we assign a zero to the tracts

– Other analysts might derive different DCF values using the same geological and cost data

Option Valuation Comparisons

• Comparison with Industry Bids– The cost and geological data used by the USGS

may deviate from industry expectations

– Even if the underlying USGS data match industry expectations, we still do not observer industry valuations directly

Option Valuation Comparisons

• Result– Compare between option valuation, USGS

and industry bid values• OV – option valuation• USGS – USGS DCF valuation• GB – Geometric mean of industry bids• HG – High (winning) industry bid

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Comparative Statics

• Variance– Given that oil and gas have been found, there is little

likelihood that exploration and development will not occur immediately

• Relinquishment requirement• Both of them do not have much effect in the data

set. But they would affect tract value in areas subject to higher unit investment cost

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Exploration and Development Timing

• Low-cost tracts should be explored or developed immediately

• High-cost tracts should be held from exploration or development

• The firm need only calculate C = V/D to decide whether a tract should be explored or developed immediately