by: marco antonio guimarães dias consultant by petrobras, brazil doctoral candidate by puc-rio...

By: Marco Antonio Guimarães Dias Consultant by Petrobras, Brazil Doctoral Candidate by PUC-Rio

Visit the first real options website: www.puc-rio.br/marco.ind/

. Overview of Real Options in Petroleum

Workshop on Real Options

Turku, Finland - May 6-8, 2002

Presentation Outline Introduction and overview of real options in upstream petroleum

(exploration & production) Intuition and classical model Stochastic processes for oil prices (with real case study)

Applications of real options in petroleum Petrobras research program called “PRAVAP-14 ” Valuation of

Development Projects under Uncertainties Combination of technical and market uncertainties in most cases

Selection of mutually exclusive alternatives for oilfield development under oil price uncertainty

Exploratory investment and information revelation Investment in information: dynamic value of information Option to expand the production with optional wells

Managerial View of Real Options (RO) RO is a modern methodology for economic evaluation of projects

and investment decisions under uncertainty RO approach complements (not substitutes) the corporate tools (yet) Corporate diffusion of RO takes time, training, and marketing

RO considers the uncertainties and the options (managerial flexibilities), giving two interconnected answers: The value of the investment opportunity (value of the option); and The optimal decision rule (threshold)

RO can be viewed as an optimization problem: Maximize the NPV (typical objective function) subject to: (a) Market uncertainties (eg.: oil price); (b) Technical uncertainties (eg., reserve volume); (c) Relevant Options (managerial flexibilities); and (d) Others firms interactions (real options + game theory)

Main Petroleum Real Options and Examples

Option to Expand the Production Depending of market scenario (oil prices, rig rates)

and the petroleum reservoir behavior, new wells can be added to the production system

Option to Delay (Timing Option) Wait, see, learn, optimize before invest Oilfield development; wildcat drilling

Abandonment Option Managers are not obligated to continue a business plan if it becomes unprofitable Sequential appraisal program can be abandoned earlier if information generated is not favorable

Undelineated Field: Option to Appraise

Appraisal InvestmentAppraisal Investment

RevisedVolume = B’

Developed Reserves: Options to Expand, to Stop Temporally, and to Abandon.

E&P as a Sequential Real Options Process Concession: Option to Drill the Wildcat

Exploratory (wildcat) Exploratory (wildcat) InvestmentInvestment

Oil/Gas SuccessProbability = p

Expected Volumeof Reserves = B

Delineated Undeveloped Reserves: Option to Develop (What is the best alternative?)

Development InvestmentDevelopment Investment

Economic Quality of the Developed Reserve Imagine that you want to buy 100 million barrels of developed oil

reserves. Suppose a long run oil price is 20 US$/bbl. How much you shall pay for each barrel of developed reserve?

It depends of many factors like the reservoir permo-porosity quality (productivity), fluids quality (heavy x light oil, etc.), country (fiscal regime, politic risk), specific reserve location (deepwaters has higher operational cost than onshore reserve), the capital in place (extraction speed and so the present value of revenue depends of number of producing wells), etc.

As higher is the percentual value for the reserve barrel in relation to the barrel oil price (on the surface), higher is the economic quality: value of one barrel of reserve = v = q . P Where q = economic quality of the developed reserve The value of the developed reserve is v times the reserve size (B) So, let us use the equation for NPV = V D = q P B D

D = development cost (investment cost or exercise price of the option)

Intuition (1): Timing Option and Oilfield Value Assume that simple equation for the oilfield development NPV:

NPV = q B P D = 0.2 x 500 x 18 – 1850 = 50 million $ Do you sell the oilfield for US$ 3 million? Suppose the following two-periods problem and uncertainty with only two scenarios at

the second period for oil prices P.

E[P] = 18 $/bblNPV(t=0) = 50 million $

P+ = 19 NPV = + 50 million $

P = 17 NPV = 150 million $ Rational manager will not exercise this option Max (NPV, 0) = zero

Hence, at t = 1, the project NPV is positive: (50% x 50) + (50% x 0) = + 25 million $

50%

50%

t = 1

t = 0

Intuition (2): Timing Option and Waiting Value Suppose the same case but with a small positive NPV. What is better: develop now

or wait and see? NPV = q B P D = 0.2 x 500 x 18 – 1750 = 50 million $ Discount rate = 10%

E[P] = 18 /bblNPV(t=0) = 50 million $

P+ = 19 NPV+ = + 150 million $

P = 17 NPV = 50 million $ Rational manager will not exercise this option Max (NPV, 0) = zero

Hence, at t = 1, the project NPV is: (50% x 150) + (50% x 0) = + 75 million $

The present value is: NPVwait(t=0) = 75/1.1 = 68.2 > 50

50%

50%

t = 1

t = 0

Hence is better to wait and see, exercising the option only in favorable scenario

Intuition (3): Deep-in-the-Money Real Option Suppose the same case but with a higher NPV.

What is better: develop now or wait and see? NPV = q B P D = 0.25 x 500 x 18 – 1750 = 500 million $ Discount rate = 10%

E[P] = 18 /bbl NPV(t=0) = 500 million $

P+ = 19 NPV = 625 million $

P = 17 NPV = 375 million $

Hence, at t = 1, the project NPV is: (50% x 625) + (50% x 375) = 500 million $

The present value is: NPVwait(t=0) = 500/1.1 = 454.5 < 500

50%

50%

t = 1

t = 0

Immediate exercise is optimal because this project is deep-in-the-money (high NPV)

Later, will be discussed the problem of probability, discount rate, etc.

When Real Options Are Valuable? Based on the textbook “Real Options” by Copeland & Antikarov

Real options are as valuable as greater are the uncertainties and the flexibility to respondA

bili

ty t

o re

spon

d

Low

High

Likelihood of receiving new informationLow High

U n c e r t a i n t y

Roo

m f

or

Man

ager

ial F

lexi

bil

ity

Moderate Flexibility Value

Moderate Flexibility Value

Low Flexibility Value

High

Flexibility Value

Classical Real Options in Petroleum Model Paddock & Siegel & Smith wrote a series of papers on

valuation of offshore reserves in 80’s (published in 87/88) It is the best known model for oilfields development decisions It explores the analogy financial options with real options Uncertainty is modeled using the Geometric Brownian Motion

Black-Scholes-Merton’s Financial Options Paddock, Siegel & Smith’s Real Options

Financial Option Value Real Option Value of an Undeveloped Reserve (F)

Current Stock Price Current Value of Developed Reserve (V)

Exercise Price of the Option Investment Cost to Develop the Reserve (D)

Stock Dividend Yield Cash Flow Net of Depletion as Proportion of V ()

Risk-Free Interest Rate Risk-Free Interest Rate (r)

Stock Volatility Volatility of Developed Reserve Value ()

Time to Expiration of the Option Time to Expiration of the Investment Rights ()

Estimating the Underlying Asset Value How to estimate the value of underlying asset V?

Transactions in the developed reserves market (USA) v = value of one barrel of developed reserve (stochastic); V = v B where B is the reserve volume (number of barrels); v is ~ proportional to petroleum prices P, that is, v = q P ; For q = 1/3 we have the one-third rule of thumb (USA mean);

– So, Paddock et al. used the concept of economic quality (q) – This is a business view on reserve value (reserves market oriented view)

Discounted cash flow (DCF) estimate of V, that is: NPV = V D V = NPV + D For fiscal regime of concessions the chart NPV x P is a straight line, so

that we can assume that V is proportional to P Again is used the concept of quality of reserve, but calculated from a

DCF spreadsheet, which everybody use in oil companies. Let us see how.

NPV x P Chart and the Quality of Reserve

tangent = q . B

D

P ($/bbl)

NP

V (

mil

lion

$) Linear Equation for the NPV:

NPV = q P B D

NPV in function of P

The quality of reserve (q) is relatedwith the inclination of the NPV line

Using a simple DCF spreadsheet we can get the reserve quality value

Estimating the Model Parameters If V = k P, we have V = P and V = P (D&P p.178. Why?)

Risk-neutral Geometric Brownian: dV = (r V) V dt + V V dz

Volatility of long-term oil prices (~ 20% p.a.) For development decisions the value of the benefit is linked to the long-term oil

prices, not the (more volatile) spot prices A good market proxy is the longest maturity contract in futures markets with

liquidity (Nymex 18th month; Brent 12th month) Volatily = standard-deviation of ( Ln Pt Ln Pt1 )

Dividend yield (or long-term convenience yield) ~ 6% p.a. Paddock & Siegel & Smith: equation using cash-flows If V = k P, we can estimate from oil prices futures market

Pickles & Smith’s Rule (1993): r = (in the long-run) “We suggest that option valuations use, initially, the ‘normal’ value of net convenience yield, which seems to

equal approximately the risk-free nominal interest rate”

NYMEX-WTI Oil Prices: Spot x Futures Note that the spot prices reach more extreme values and have more

‘nervous’ movements (more volatile) than the long-term futures pricesWTI Nymex Prices: Spot (First Month) vs. 18 Months

Jul/1996 - Jan/2002

5

10

15

20

25

30

35

407/

22/1

996

10/2

2/19

96

1/22

/199

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/199

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97

1/22

/199

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/200

2

WT

I (U

S$/

bb

l)

WTI Nymex Spot (1st Mth) Close (US$/bbl)

WTI Nymex Mth18 Close (US$/bbl)

Equation of the Undeveloped Reserve (F) Partial (t, V) Differential Equation (PDE) for the option F

Managerial Action Is Inserted into the Model

}} Conditions at the Point of Optimal Early Investment

Conditions at the Point of Optimal Early Investment

Boundary Conditions:

For V = 0, F (0, t) = 0 For t = T, F (V, T) = max [V D, 0] = max [NPV, 0] For V = V*, F (V*, t) = V* D “Smooth Pasting”, FV (V*, t) = 1

Parameters: V = value of developed reserve (eg., V = q P B); D = development cost; r = risk-free discount rate; = dividend yield for V ; = volatility of V

0.5 2 V2 FVV + (r ) V FV r F = Ft

The Undeveloped Oilfield Value: Real Options and NPV Assume that V = q B P, so that we can use chart F x V or F x P

Suppose the development break-even (NPV = 0) occurs at US$15/bbl

Threshold Curve: The Optimal Decision Rule At or above the threshold line, is optimal the immediate

development. Below the line: “wait, learn and see”

Stochastic Processes for Oil Prices: GBM Like Black-Scholes-Merton equation, the classic model of Paddock et al uses the popular Geometric

Brownian Motion Prices have a log-normal distribution in every future time; Expected curve is a exponential growth (or decline); In this model the variance grows with the time horizon

In this process, the price tends to revert towards a long-run average price (or an equilibrium level) P. Model analogy: spring (reversion force is proportional to the distance between current position and the

equilibrium level). In this case, variance initially grows and stabilize afterwards

Mean-Reverting Process

Stochastic Processes Alternatives for Oil Prices There are many models of stochastic processes for oil prices in real

options literature. I classify them into three classes.

The nice properties of Geometric Brownian Motion (few parameters, homogeneity) is a great incentive to use it in real options applications. Pindyck (1999) wrote: “the GBM assumption is unlikely to lead to large errors in the optimal investment rule”

Mean-Reversion + Jump: the Sample Paths 100 sample paths for mean-reversion + jumps ( = jump each 5years)

Nominal Prices for Brent and Similar Oils (1970-2001) With an adequate long-term scale, we can see that oil prices jump in both directions, depending of the kind

of abnormal news: jumps-up in 1973/4, 1978/9, 1990, 1999; and jumps-down in 1986, 1991, 1997, 2001

Jumps-up Jumps-down

Mean-Reversion + Jumps: Dias & Rocha We (Dias & Rocha, 1998/9) adapt the Merton (1976) jump-diffusion idea for the oil prices

case, considering: Normal news cause only marginal adjustment in oil prices, modeled with the continuous-time process of

mean-reversion Abnormal rare news (war, OPEC surprises, ...) cause abnormal adjustment (jumps) in petroleum prices,

modeled with a discrete-time Poisson process (we allow both jumps-up & jumps-down) Model has more economic logic (supply x demand)

Normal information causes smoothing changes in oil prices (marginal variations) and means both: Marginal interaction between production and demand (inventory levels as indicator); and Depletion versus new reserves discoveries for non-OPEC (the ratio of reserves/production is an indicator)

Abnormal information means very important news: In few months, this kind of news causes jumps in the prices, due to large variation (or expected large variation) in either

supply or demand

Real Case with Mean-Reversion + Jumps A similar process of mean-reversion with jumps was used by Dias

for the equity design (US$ 200 million) of the Project Finance of Marlim Field (oil prices-linked spread) Equity investors reward:

Basic interest-rate + (oil business risk linked) spread Oil prices-linked: transparent deal (no agency cost) and win-win:

Higher oil prices higher spread, and vice versa (good for both)

Deal was in December 1998 when oil price was 10 $/bbl We convince investors that the expected oil prices curve was a fast reversion

towards US$ 20/bbl (equilibrium level) Looking the jumps-up & down, we limit the spread by putting both cap

(maximum spread) and floor (to prevent negative spread) This jumps insight proved be very important:

Few months later the oil prices jump-up (price doubled by Aug/99)– The cap protected Petrobras from paying a very high spread

PRAVAP-14: Some Real Options Projects PRAVAP-14 is a systemic research program named Valuation

of Development Projects under Uncertainties I coordinate this systemic project by Petrobras/E&P-Corporative

I’ll present some real options projects developed: Selection of mutually exclusive alternatives of development investment under

oil prices uncertainty (with PUC-Rio) Exploratory revelation with focus in bids (pre-PRAVAP-14) Dynamic value of information for development projects Analysis of alternatives of development with option to expand, considering

both oil price and technical uncertainties (with PUC) We analyze different stochastic processes and solution methods

Geometric Brownian, reversion + jumps, different mean-reversion models Finite differences, Monte Carlo for American options, genetic algorithms Genetic algorithms are used for optimization (thresholds curves evolution)

I call this method of evolutionary real options (I have two papers on this)

E&P Process and Options Drill the wildcat (pioneer)? Wait and See? Revelation: additional waiting incentives



RevisedVolume = B’ Appraisal phase: delineation of reserves

Invest in additional information?

Delineated but Undeveloped Reserves. Develop? “Wait and See” for better

conditions? What is the best alternative?

Developed Reserves. Expand the production?

Stop Temporally? Abandon?

Selection of Alternatives under Uncertainty In the equation for the developed reserve value V = q P B, the

economic quality of reserve (q) gives also an idea of how fast the reserve volume will be produced. For a given reserve, if we drill more wells the reserve will be depleted faster,

increasing the present value of revenues Higher number of wells higher q higher V However, higher number of wells higher development cost D

For the equation NPV = q P B D, there is a trade off between q and D, when selecting the system capacity (number of wells, the platform process capacity, pipeline diameter, etc.) For the alternative “j” with n wells, we get NPVj = qj P B Dj

Hence, an important investment decision is: How select the best one from a set of mutually exclusive alternatives? Or, What is the

best intensity of investment for a specific oilfield? I follow the paper of Dixit (1993), but considering finite-lived options.

The Best Alternative at Expiration (Now or Never) The chart below presents the “now-or-never” case for three alternatives. In this case, the NPV

rule holds (choose the higher one). Alternatives: A1(D1, q1); A2(D1, q1); A3(D3, q3), with D1 < D2 < D3 and q1 < q2 < q3

Hence, the best alternative depends on the oil price P. However, P is uncertain!

The Best Alternative Before the Expiration Imagine that we have years before the expiration and in addition

the long-run oil prices follow the geometric Brownian We can calculate the option curves for the three alternatives, drawing only

the upper real option curve(s) (in this case only A2), see below.

The decision rule is: If P < P*2 , “wait and see”

Alone, A1 can be even deep-in-the-money, but wait for A2 is more valuable

If P = P*2 , invest now with A2

Wait is not more valuable

If P > P*2 , invest now with the higher NPV alternative (A2 or A3 ) Depending of P, exercise A2 or A3

How about the decision rule along the time? (thresholds curve) Let us see from a PRAVAP-14 software

Threshold Curves for Three Alternatives There are regions of wait and see and others that the immediate investment is optimal for

each alternative

InvestmentsD3 > D2 > D1






Delineated but Undeveloped Reserves. Develop? “Wait and See” for better conditions? What is the best

alternative?



Technical Uncertainty and Risk Reduction Technical uncertainty decreases when efficient investments in information are performed ( learning process). Suppose a new basin with large geological uncertainty. It is reduced by the exploratory investment of the whole industry

The “cone of uncertainty” (Amram & Kulatilaka) can be adapted to understand the technical uncertainty:

Risk reduction by the investment in information of all firms in the basin(driver is the investment, not the passage of time directly)

Project evaluation with additionalinformation(t = T)

Lower Risk

ExpectedValue

Current project evaluation(t=0)

HigherRisk

ExpectedValue

con

fid

ence

in

terv

al

Lack of Knowledge Trunk of Cone

Technical Uncertainty and Revelation But in addition to the risk reduction process, there is another important issue:

revision of expectations (revelation process) The expected value after the investment in information (conditional expectation) can be very

different of the initial estimative Investments in information can reveal good or bad news

Value withgood revelation

Value withbad revelation

Current project evaluation (t=0)

Investment inInformation

Project valueafter investment

t = T

Value withneutral revelation

E[V]

Technical Uncertainty in New Basins The number of possible scenarios to be revealed (new expectations) is proportional to the cumulative investment in

information Information can be costly (our investment) or free, from the other firms investment (free-rider) in this under-explored basin

The arrival of information process leverage the option value of a tract

.

Investmentin information(wildcat drilling, etc.)

Investment in information(costly and free-rider)

Todaytechnicaland economicvaluation

t = 0 t = 1

Possible scenariosafter the informationarrived during the first year of option term

t = T

Possible scenariosafter the informationarrived during the option leaseterm

RevelationDistribution

Valuation of Exploratory Prospect Suppose that the firm has 5 years option to drill the wildcat

Other firm wants to buy the rights of the tract for $ 3 million $. Do you sell? How valuable is the prospect?

E[B] = 150 million barrels (expected reserve size)

E[q] = 20% (expected quality of developed reserve)

P(t = 0) = US$ 20/bbl (long-run expected price at t = 0)

D(B) = 200 + (2 . B) D(E[B]) = 500 million $

NPV = q P B D = (20% . 20 . 150) 500 = + 100 MM$ However, there is only 15% chances to find petroleum

EMV = Expected Monetary Value = IW + (CF . NPV) EMV = 20 + (15% . 100) = 5 million $

20 million $(IW = wildcat investment)

15% (CF = chance factor)

Dry Hole

“Compact Tree”

Su

cces

s

Do you sell the prospect rights for US$ 3 million?

Monte Carlo Combination of Uncertainties Considering that: (a) there are a lot of uncertainties in that low known basin; and (b) many oil companies will drill wildcats in that area in the

next 5 years: The expectations in 5 years almost surely will change and so the prospect value The revelation distributions and the risk-neutral distribution for oil prices are:

Dis

trib

utio

n o

f E

xpec

tati

ons

(Rev

elat

ion

Dis

trib

uti

ons)

Real x Risk-Neutral Simulation The GBM simulation paths: one real () and the other risk-neutral (r ). In reality r = , where is a

risk-premium

0

5

10

15

20

25

30

35

40

45

0.0

0.3

0.5

0.8

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2.5

2.8

3.0

3.3

3.5

3.8

4.0

4.3

4.5

4.8

5.0

5.3

5.5

5.8

6.0

Time (Years)

Oil

Pri

ce ($

/bb

l)

Real Simulation

Risk-Neutral Simulation

A Visual Equation for Real Options

Prospect Evaluation(in million $)

Traditional Value = 5

Options Value (at T) = + 12.5

Options Value (at t=0) = + 7.6

+

Today the prospect´s EMV is negative, but there is 5 years for wildcat decision and new scenarios will be revealed by the exploratory investment in that basin.

=So, refuse the $ 3 million offer!






Delineated but Undeveloped Reserves. Develop? “Wait and See” for better conditions? What is the best

alternative?



A Dynamic View on Value of Information Value of Information has been studied by decision analysis theory. I extend this view

using real options tools, adopting the name dynamic value of information. Why dynamic? Because the model takes into account the factor time:

Time to expiration for the real option to commit the development plan; Time to learn: the learning process takes time. Time of gathering data, processing, and analysis to get new

knowledge on technical parameters Continuous-time process for the market uncertainties (oil prices) interacting with the current expectations of

technical parameters

How to model the technical uncertainty and its evolution after one or more investment in information? The process of accumulating data about a technical parameter is a learning process towards the

“truth” about this parameter This suggest the names of information revelation and revelation distribution

In finance (even in derivatives) we work with expectations Revelation distribution is the distribution of conditional expectations

– The conditioning is the new information (see details in www.realoptions.org/)

Simulation Issues The differences between the oil prices and revelation processes are:

Oil price (and other market uncertainties) evolves continually along the time and it is non-controllable by oil companies (non-OPEC)

Revelation distributions occur as result of events (investment in information) in discrete points along the time In many cases (appraisal phase) only our investment in information is

relevant and it is totally controllable by us (activated by management)

P

E[B]Inv

Inv

Let us consider that the exercise price of the option (development cost D) is function of B. So, D changes just at the information revelation on B. In order to calculate only one development threshold we work with the

normalized threshold (V/D)* that doesn´t change in the simulation

Combination of Uncertainties in Real Options The Vt/D sample paths are checked with the threshold (V/D)*

A

Option F(t = 1) = V DF(t = 0) == F(t=1) * exp (r*t)

Present Value (t = 0)

B

F(t = 2) = 0ExpiredWorthless

Vt/D = (q Pt B)/D(B)

E&P Process and Options Drill the wildcat? Wait? Extend? Revelation, option-game: waiting incentives




Technical uncertainty: sequential options

Developed Reserves. Expand the production? Stop Temporally? Abandon?

Delineated but Undeveloped Reserves. Develop? Wait and See? Extend the option? Invest in additional

information?

Option to Expand the Production Analyzing a large ultra-deepwater project in Campos Basin, Brazil,

we faced two problems: Remaining technical uncertainty of reservoirs is still important.

In this specific case, the best way to solve the uncertainty is not by drilling additional appraisal wells. It’s better learn from the initial production profile.

In the preliminary development plan, some wells presented both reservoir risk and small NPV. Some wells with small positive NPV (are not “deep-in-the-money”) Depending of the information from the initial production, some wells could be not

necessary or could be placed at the wrong location.

Solution: leave these wells as optional wells Buy flexibility with an additional investment in the production system:

platform with capacity to expand (free area and load) It permits a fast and low cost future integration of these wells

The exercise of the option to drill the additional wells will depend of both market (oil prices, rig costs) and the initial reservoir production response

Oilfield Development with Option to Expand The timeline below represents a case analyzed in PUC-Rio

project, with time to build of 3 years and information revelation with 1 year of accumulated production

The practical “now-or-never” is mainly because in many cases the effect of secondary depletion is relevant The oil migrates from the original area so that the exercise of the option gradually become less probable (decreasing

NPV) In addition, distant exercise of the option has small present value Recall the expenses to embed flexibility occur between t = 0 and t = 3

petroleum reservoir (top view) and the grid of wells

Secondary Depletion Effect: A Complication With the main area production, occurs a slow oil migration from the optional

wells areas toward the depleted main area

optional wells

oil migration (secondary depletion)

It is like an additional opportunity cost to delay the exercise of the option to expand. So, the effect of secondary depletion is like the effect of dividend yield

Modeling the Option to Expand Define the quantity of wells “deep-in-the-money” to start the basic

investment in development Define the maximum number of optional wells Define the timing (accumulated production) that reservoir

information will be revealed and the revelation distributions Define for each revealed scenario the marginal production of each

optional well as function of time. Consider the secondary depletion if we wait after learn about reservoir

Add market uncertainty (stochastic process for oil prices) Combine uncertainties using Monte Carlo simulation Use an optimization method to consider the earlier exercise of the

option to drill the wells, and calculate option value Monte Carlo for American options is a growing research area Many Petrobras-PUC projects use Monte Carlo for American options

Conclusions The real options models in petroleum bring a rich framework to

consider optimal investment under uncertainty, recognizing the managerial flexibilities Traditional discounted cash flow is very limited and can induce to

serious errors in negotiations and decisions We saw the classical model, working with the intuition

We saw different stochastic processes and other models I gave an idea about the real options research at Petrobras and

PUC-Rio (PRAVAP-14) We saw options along all petroleum E&P process We worked mainly with models combining technical uncertainties with

market uncertainty (Monte Carlo for American options) The model using the revelation distribution gives the correct incentives for investment

in information (more formal paper in Cyprus, July 2002)

Thank you very much for your time

Anexos

APPENDIXSUPPORT SLIDES

See more on real options in the first website on real options at: http://www.puc-rio.br/marco.ind/

Example in E&P with the Options Lens In a negotiation, important mistakes can be done if we don´t

consider the relevant options Consider two marginal oilfields, with 100 million bbl, both non-

developed and both with NPV = 3 millions in the current market conditions The oilfield A has a time to expiration for the rights of only 6 months, while for

the oilfield B this time is of 3 years Cia X offers US 1 million for the rights of each oilfield. Do you

accept the offer? With the static NPV, these fields have no value and even worse, we

cannot see differences between these two fields It is intuitive that these rights have value due the uncertainty and the option to wait

for better conditions. Today the NPV is negative, but there are probabilities for the NPV become positive in the future

In addition, the field B is more valuable (higher option) than the field A

Other Parameters for the Simulation Other important parameters are the risk-free interest rate r

and the dividend yield (or convenience yield for commodities) Even more important is the difference r (the risk-neutral drift) or

the relative value between r and Pickles & Smith (Energy Journal, 1993) suggest for long-run

analysis (real options) to set r = “We suggest that option valuations use, initially, the ‘normal’ value of , which

seems to equal approximately the risk-free nominal interest rate. Variations in this value could then be used to investigate sensitivity to parameter changes induced by short-term market fluctuations”

Reasonable values for r and range from 4 to 8% p.a. By using r = the risk-neutral drift is zero, which looks reasonable

for a risk-neutral process

Relevance of the Revelation Distribution Investments in information permit both a reduction of the uncertainty and a revision of our expectations on the basic technical parameters.

Firms use the new expectation to calculate the NPV or the real options exercise payoff. This new expectation is conditional to information. When we are evaluating the investment in information, the conditional expectation of the parameter X is itself a random variable E[X | I] The distribution of conditional expectations E[X | I] is named here revelation distribution, that is, the distribution of RX = E[X | I]

The concept of conditional expectation is also theoretically sound: We want to estimate X by observing I, using a function g( I ). The most frequent measure of quality of a predictor g is its mean square error defined by MSE(g) = E[X g( I )]2 . The choice of g* that minimizes the error measure MSE(g) is exactly the conditional expectation E[X | I ]. This is a very known property used in econometrics

The revelation distribution has nice practical properties (propositions)

The Revelation Distribution Properties Full revelation definition: when new information reveal all the truth about the technical parameter, we have full revelation

Much more common is the partial revelation case, but full revelation is important as the limit goal for any investment in information process The revelation distributions RX (or distributions of conditional expectations with the new information) have at least 4 nice properties for the real options practitioner:

Proposition 1: for the full revelation case, the distribution of revelation RX is equal to the unconditional (prior) distribution of X Proposition 2: The expected value for the revelation distribution is equal the expected value of the original (a priori) technical parameter X distribution

That is: E[E[X | I ]] = E[RX] = E[X] (known as law of iterated expectations) Proposition 3: the variance of the revelation distribution is equal to the expected reduction of variance induced by the new information

Var[E[X | I ]] = Var[RX] = Var[X] E[Var[X | I ]] = Expected Variance Reduction Proposition 4: In a sequential investment process, the ex-ante sequential revelation distributions {RX,1, RX,2, RX,3, …} are (event-driven) martingales

In short, ex-ante these random variables have the same mean

Investment in Information x Revelation Propositions Suppose the following stylized case of investment in information in order to get

intuition on the propositions Only one well was drilled, proving 100 MM bbl (MM = million)

A B

DC

Area A: provedBA = 100 MM bbl

Area B: possible50% chances of

BB = 100 MM bbl& 50% of nothing

Area D: possible50% chances of

BD = 100 MM bbl& 50% of nothing

Area C: possible50% chances of

BC = 100 MM bbl& 50% of nothing

Suppose there are three alternatives of investment in information (with different revelation powers): (1) drill one well (area B); (2) drill two wells (areas B + C); (3) drill three wells (B + C + D)

Alternative 0 and the Total Technical Uncertainty Alternative Zero: Not invest in information

This case there is only a single scenario, the current expectation So, we run economics with the expected value for the reserve B:

E(B) = 100 + (0.5 x 100) + (0.5 x 100) + (0.5 x 100)

E(B) = 250 MM bbl But the true value of B can be as low as 100 and as higher as 400 MM

bbl. Hence, the total uncertainty is large. Without learning, after the development you find one of the values:

100 MM bbl with 12.5 % chances (= 0.5 3 ) 200 MM bbl with 37,5 % chances (= 3 x 0.5 3 ) 300 MM bbl with 37,5 % chances 400 MM bbl with 12,5 % chances

The variance of this prior distribution is 7500 (million bbl)2

Alternative 1: Invest in Information with Only One Well Suppose that we drill only the well in the area B.

This case generated 2 scenarios, because the well B result can be either dry (50% chances) or success proving more 100 MM bbl

In case of positive revelation (50% chances) the expected value is:

E1[B|A1] = 100 + 100 + (0.5 x 100) + (0.5 x 100) = 300 MM bbl In case of negative revelation (50% chances) the expected value is:

E2[B|A1] = 100 + 0 + (0.5 x 100) + (0.5 x 100) = 200 MM bbl Note that with the alternative 1 is impossible to reach extreme scenarios like 100 MM bbl or

400 MM bbl (its revelation power is not sufficient)

So, the expected value of the revelation distribution is: EA1[RB] = 50% x E1(B|A1) + 50% x E2(B|A1) = 250 million bbl = E[B]

As expected by Proposition 2

And the variance of the revealed scenarios is: VarA1[RB] = 50% x (300 250)2 + 50% x (200 250)2 = 2500 (MM bbl)2

Let us check if the Proposition 3 was satisfied

Alternative 1: Invest in Information with Only One Well In order to check the Proposition 3, we need to calculated the expected

reduction of variance with the alternative A1

The prior variance was calculated before (7500). The posterior variance has two cases for the well B outcome:

In case of success in B, the residual uncertainty in this scenario is: 200 MM bbl with 25 % chances (in case of no oil in C and D) 300 MM bbl with 50 % chances (in case of oil in C or D) 400 MM bbl with 25 % chances (in case of oil in C and D)

The negative revelation case is analog: can occur 100 MM bbl (25% chances); 200 MM bbl (50%); and 300 MM bbl (25%)

The residual variance in both scenarios are 5000 (MM bbl)2

So, the expected variance of posterior distribution is also 5000 So, the expected reduction of uncertainty with the alternative A1 is: 7500 –

5000 = 2500 (MM bbl)2

Equal variance of revelation distribution(!), as expected by Proposition 3

Visualization of Revealed Scenarios: Revelation Distribution

This is exactly the prior distribution of B (Prop. 1 OK!)

All th

e revelation d

istribu

tions h

ave the sam

e mean

(marin

gale): Prop

. 4 OK

!

Posterior Distribution x Revelation Distribution The picture below help us to answer the question: Why learn?

Reduction of technical uncertainty

Increase thevariance ofrevelationdistribution(and so the option value)

Why learn?

Revelation Distribution and the Experts The propositions allow a practical way to ask the technical expert on the

revelation power of any specific investment in information. It is necessary to ask him/her only 2 questions: What is the total uncertainty on each relevant technical parameter? That is, the

probability distribution (and its mean and variance). By proposition 1, the variance of total initial uncertainty is the variance limit for the revelation

distribution generated from any investment in information By proposition 2, the revelation distribution from any investment in information has the same

mean of the total technical uncertainty.

For each alternative of investment in information, what is the expected reduction of variance on each technical parameter? By proposition 3, this is also the variance of the revelation distribution

In addition, the discounted cash flow analyst together with the reservoir engineer, need to find the penalty factor up: Without full information about the size and productivity of the reserve, the non-

optimized system doesn´t permit to get the full project value

Non-Optimized System and Penalty Factor If the reserve is larger (and/or more productive) than expected, with the

limited process plant capacity the reserves will be produced slowly than in case of full information. This factor can be estimated by running a reservoir simulation with limited process

capacity and calculating the present value of V.

The NPV with technical uncertainty is calculated using Monte Carlo simulation and the equations:

NPV = q P B D(B) if q B = E[q B]

NPV = q P B up D(B) if q B > E[q B]

NPV = q P B down D(B) if q B < E[q B]

In general we have down = 1 and up < 1

The Normalized Threshold and Valuation Recall that the development option is optimally exercised at the

threshold V*, when V is suficiently higher than D Exercise the option only if the project is “deep-in-the-money”

Assume D as a function of B but approximately independent of q. Assume the linear equation: D = 310 + (2.1 x B) (MM$)

This means that if B varies, the exercise price D of our option also varies, and so the threshold V*. The computational time for V* is much higher than for D

We need perform a Monte Carlo simulation to combine the uncertainties after an information revelation. After each B sampling, it is necessary to calculate the new threshold curve

V*(t) to see if the project value V = q P B is deep-in-the money In order to reduce the computational time, we work with the

normalized threshold (V/D)*. Why?

Normalized Threshold and Valuation We will perform the valuation considering the optimal exercise at the

normalized threshold level (V/D)* After each Monte Carlo simulation combining the revelation distributions of q and B

with the risk-neutral simulation of P We calculate V = q P B and D(B), so V/D, and compare with (V/D)*

Advantage: (V/D)* is homogeneous of degree 0 in V and D. This means that the rule (V/D)* remains valid for any V and D So, for any revealed scenario of B, changing D, the rule (V/D)* remains This was proved only for geometric Brownian motions (V/D)*(t) changes only if the risk-neutral stochastic process parameters r, , change.

But these factors don’t change at Monte Carlo simulation

The computational time of using (V/D)* is much lower than V* The vector (V/D)*(t) is calculated only once, whereas V*(t) needs be re-calculated

every iteration in the Monte Carlo simulation. In addition V* is a time-consuming calculus

Overall x Phased Development Let us consider two alternatives

Overall development has higher NPV due to the gain of scale Phased development has higher capacity to use the information along

the time, but lower NPV With the information revelation from Phase 1, we can

optimize the project for the Phase 2 In addition, depending of the oil price scenario and other market and

technical conditions, we can not exercise the Phase 2 option The oil prices can change the decision for Phased development, but not

for the Overall development alternativeThe valuation is similar to the previously presented

Only by running the simulations is possible to compare the higher NPV versus higher flexibility

Real Options Evaluation by Simulation + Threshold Curve Before the information revelation, V/D changes due the oil prices P (recall V = qPB and NPV = V – D). With

revelation on q and B, the value V jumps.

A

Option F(t = 5.5) = V DF(t = 0) == F(t=5.5) * exp (r*t)

Present Value (t = 0)

B

F(t = 8) = 0Expires Worthless

Oil Drilling Bayesian Game (Dias, 1997) Oil exploration: with two or few oil companies exploring a basin, can be important to consider the waiting game of drilling Two companies X and Y with neighbor tracts and correlated oil prospects: drilling reveal information

If Y drills and the oilfield is discovered, the success probability for X’s prospect increases dramatically. If Y drilling gets a dry hole, this information is also valuable for X.

In this case the effect of the competitor presence is to increase the value of waiting to invest

Company X tractCompany X tract Company Y tractCompany Y tract

Two Sequential Learning: Schematic Tree Two sequential investment in information (wells “B” and “C”):

InvestWell “B”

RevelationScenarios

PosteriorScenarios

InvestWell “C”

50%

50%

50%

50%

50%

50%

{ 400300

{ 300200

{ 200100

350 (with 25% chances)

The upper branch means good news, whereas the lower one means bad news



NPV

300

100

- 200

Visual FAQ’s on Real Options: 9 Is possible real options theory to recommend

investment in a negative NPV project?

Answer: yes, mainly sequential options with investment revealing new informations Example: exploratory oil prospect (Dias 1997)

Suppose a “now or never” option to drill a wildcatStatic NPV is negative and traditional theory recommends to give up the

rights on the tractReal options will recommend to start the sequential investment, and

depending of the information revealed, go ahead (exercise more options) or stop

Sequential Options (Dias, 1997)

Traditional method, looking only expected values, undervaluate the prospect (EMV = 5 MM US$): There are sequential options, not sequential obligations; There are uncertainties, not a single scenario.

( Wildcat Investment )

( Developed Reserves Value )

( Appraisal Investment: 3 wells )

( Development Investment )

Note: in million US$“Compact Decision-Tree”

EMV = 15 + [20% x (400 50 300)] EMV = 5 MM$

Sequential Options and Uncertainty Suppose that each appraisal

well reveal 2 scenarios (good and bad news)

development option will not be exercised by rational managers

option to continue the appraisal phase will not be exercised by rational managers

Option to Abandon the Project Assume it is a “now or never” option If we get continuous bad news, is better to stop

investment Sequential options turns the EMV to a positive

value The EMV gain is

3.25 5) = $ 8.25 being:

(Values in millions)

$ 2.25 stopping development

$ 6 stopping appraisal

$ 8.25 total EMV gain

Economic Quality of a Developed Reserve Concept by Dias (1998): q = v/P ; v = V/B (in $/bbl)

q = economic quality of the developed reserve v = value of one barrel of the developed reserve ($/bbl) P = current petroleum price ($/bbl)

For the proportional model, v = q P, the economic quality of the reserve is constant. We adopt this model. The option charts F x V and F x P at the expiration (t = T)

F

VD

45o

tg 45o = 1

F

PD/qB

(tg = q = economic quality

v = q . P; V = v . BF(t=T) = max (q P B D, 0)

F(t=T) = max (NPV, 0)NPV = V D

Monte Carlo Simulation of Uncertainties Simulation will combine uncertainties (technical and market) for the

equation of option exercise: NPV(t)dyn = q . B . P(t) D(B)

Reserve Size (B) (only at t = trevelation)

(in million of barrels)

Minimum = 300Most Likely = 500Maximum = 700

Oil Price (P) ($/bbl) (from t = 0 until t = T)

Mean = 18 US$/bbl Standard-Deviation:

changes with the time

Parameter Distribution Values (example)

Economic Quality of the Developed Reserve (q) (only at t = trevelation)

Minimum = 10% Most Likely = 15% Maximum = 20%

In the case of oil price (P) is performed a risk-neutral simulation of its stochastic process, because P(t) fluctuates continually along the time

Brent Oil Prices: Spot x Futures Note that the spot prices reach more extreme values than the long-term

futures pricesBrent Prices: Spot (Dated) vs. IPE 12 Month

Jul/1996 - Jan/2002

5

10

15

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25

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7/22

/199

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10/2

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1/22

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1/22

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2

Bre

nt

(US

$/b

bl)

Brent Platt's Dated Mid (US$/bbl)

Brent IPE Mth12 Close (US$/bbl)

Mean-Reversion + Jumps for Oil Prices Adopted in the Marlim Project Finance (equity

modeling) a mean-reverting process with jumps:

The jump size/direction are random: ~ 2N

In case of jump-up, prices are expected to double OBS: E()up = ln2 = 0.6931

In case of jump-down, prices are expected to halve OBS: ln(½) = ln2 = 0.6931

where:(the probability of jumps)

(jump size)

Equation for Mean-Reversion + Jumps The interpretation of the jump-reversion equation is:

mean-reversion drift:positive drift if P < Pnegative drift if P > P

{uncertainty fromthe continuous-timeprocess (reversion){variation of the

stochastic variablefor time interval dt

uncertainty fromthe discrete-timeprocess (jumps)

continuous (diffusion) process

discreteprocess(jumps)

by: marco antonio guimarães dias consultant by petrobras, brazil doctoral candidate by puc-rio...

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