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
Relinquish?
NO
Exploration
OPTION #1
YES
Relinquish?
Results
favorable?
NO
YES
NO
OPTION #2
Development
Extraction
YES
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 E , and receive the expected value of undeveloped
reserves
where
X * (V) 
 QX(V,T  t;D(Q))dF(Q)
Q =random
 quantity of recoverable hydrocarbons in the tract
D(Q)=per unit development cost, a function of quantity
value of a unit of developed hydrocarbon reserves
V =current
F(Q) =probability distribution over the quantity of hydrocarbons
 X(V,T  t;D(Q))=current per unit value of undeveloped reserves given the current per unit




value of a developed reserve and per unit development cost
t =current date
T =expiration date
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
Rt dt /BtVt   * dt    dz
where
B t =the number of units of petroleum in a developed reserve
Vt =the value of a unit of developed reserve
R =the instantaneous per unit time net payoff from holding the reserve
t

 * =the required rate of return to the owner
  =the instantaneous per unit time standard deviation of the rate of return
dz =an increment to diffusion process
Tract Valuation by the Option Valuation
Approach
Valuation-Petroleum Reserve Market Equilibrium
R t comes from two sources
(1)
(2)
The profits from production
The capital gain on holding the remaining petroleum
Assume a developed reserve follow an exponential decline
Then the net payoff can be written as
dBt  Bt dt
where the net payoff is over a short interval dt.
Operation profit from selling a unit of petroleum
Pt is the after-tax
Rt 
dt  Bt Pt dt  (1  dt)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
dV
 ( *   )dt    dz
V
   dt    dz
where
    *  t
t   Pt  Vt  /Vt
t =the payout rate of the producing developed reserve
 =the expected rate of capital gain


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
•
•
•
•
•
• Value of developed reserve
discounted for developed lag
Variance of rate of return • Variance of rate of change of the
value of a developed reserve
• Per unit development cost
Exercise price
• Relinquishment requirement
Time to expiration
• Riskless rate of interest
Riskless rate of interest
• Net production revenue less depletion
Dividend
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 
Boundary conditions
• X(Vt, T-t, D) = Vt – D if Ct = Ct* and Cs<Cs* for all s<t
– C t = 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
• Or paying
• Value of unexplored tract is
and get
and getting
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 j
b= 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: 2=0.02019 -> =0.142
To account for increase in perceived uncertainty:
(Jacoby and Paddock[1983])
2=0.0625 -> =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 tract
Aj= 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



 2
 gj2
2
2
D j  A j 6Qoj  Qgj   18 oj 
 6 ogj A j  ( 1)6Qoj  Qgj 



2
= 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: ogi= 0.5 oj oj
Dj
Solve for Aj to get: A 
j



 2
 gj2
2
6Q

Q

18



3



(

1)
6Q

Q
 oj gj   oj
 oj gj 
oj gj 

2

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)

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