By: Marco Antonio Guimar

1
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

2
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

3
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)

4
Main Petroleum Real Options and Examples
5
EP as a Sequential Real Options Process
6
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)

7
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.

P 19 ? NPV 50 million
EP 18 /bbl NPV(t0) - 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
8
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

P 19 ? NPV 150 million
EP 18 /bbl NPV(t0) 50 million
Hence, at t 1, the project NPV is (50 x 150)
(50 x 0) 75 million The present value
is NPVwait(t0) 75/1.1 68.2 gt 50
Hence is better to wait and see, exercising the
option only in favorable scenario
9
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

P 19 ? NPV 625 million
EP 18 /bbl NPV(t0) 500 million
Hence, at t 1, the project NPV is (50 x 625)
(50 x 375) 500 million The present value
is NPVwait(t0) 500/1.1 454.5 lt 500
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.
10
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 respond

11
Classical Real Options in Petroleum Model

• Paddock Siegel Smith wrote a series of papers
  on valuation of offshore reserves in 80s
  (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

Time to Expiration of the Option Time to
Expiration of the Investment Rights (t)
12
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.

13
NPV x P Chart and the Quality of Reserve

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

Linear Equation for the NPV NPV q P B - D
The quality of reserve (q) is relatedwith the
inclination of the NPV line
14
Estimating the Model Parameters

• If V k P, we have sV sP and dV dP (DP
  p.178. Why?)
• Risk-neutral Geometric Brownian dV (r - dV) V
  dt sV 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
  Pt-1 )
• Dividend yield (or long-term convenience yield)
  6 p.a.
• Paddock Siegel Smith equation using
  cash-flows
• If V k P, we can estimate d from oil prices
  futures market
• Pickles Smiths Rule (1993) r d (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

15
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 prices

16
Equation of the Undeveloped Reserve (F)

• Partial (t, V) Differential Equation (PDE) for
  the option F

• 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

17
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 US15/bbl

18
Threshold Curve The Optimal Decision Rule

• At or above the threshold line, is optimal the
  immediate development. Below the line wait,
  learn and see

19
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

20
Mean-Reverting Process

• 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

21
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

22
Mean-Reversion Jump the Sample Paths

• 100 sample paths for mean-reversion jumps (l
  1 jump each 5 years)

23
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
24
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

25
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

26
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/EP-Corporative
• Ill 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)

27
EP Process and Options
Oil/Gas Success Probability p

• Drill the wildcat (pioneer)? Wait and See?
• Revelation additional waiting incentives

Expected Volume of Reserves B
Revised Volume 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?

28
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.

29
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 lt D2 lt D3 and q1 lt q2 lt q3

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

30
The Best Alternative Before the Expiration

• Imagine that we have t 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 lt P2 , wait and see
• Alone, A1 can be even deep-in-the-money, but wait
  for A2 is more valuable
• If P P2 , invest now with A2
• Wait is not more valuable
• If P gt P2 , 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

31
Threshold Curves for Three Alternatives

• There are regions of wait and see and others that
  the immediate investment is optimal for each
  alternative

32
EP Process and Options
Oil/Gas Success Probability p

• Drill the wildcat (pioneer)? Wait and See?
• Revelation additional waiting incentives

Expected Volume of Reserves B
Revised Volume 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?

33
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

34
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

35
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

36
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?

? 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
Do you sell the prospect rights for US 3 million?
37
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

38
Real x Risk-Neutral Simulation

• The GBM simulation paths one real (a) and the
  other risk-neutral (r - d). In reality r - d a
  - p, where p is a risk-premium

39
A Visual Equation for Real Options

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

Prospect Evaluation (in million ) Traditional
Value - 5 Options Value (at T) 12.5
Options Value (at t0) 7.6
So, refuse the 3 million offer!
40
EP Process and Options
Oil/Gas Success Probability p

• Drill the wildcat (pioneer)? Wait and See?
• Revelation additional waiting incentives

Expected Volume of Reserves B
Revised Volume 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?

41
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/)

42
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)

• 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 doesnt change in the simulation

43
Combination of Uncertainties in Real Options

• The Vt/D sample paths are checked with the
  threshold (V/D)

Vt/D (q Pt B)/D(B)
44
EP Process and Options
Oil/Gas Success Probability p

• Drill the wildcat? Wait? Extend?
• Revelation, option-game waiting incentives

Expected Volume of Reserves B
Revised Volume B

• Appraisal phase delineation of reserves
• Technical uncertainty sequential options

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

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

45
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. Its 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

46
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

47
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

• 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

48
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

49
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 EP 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

50
Anexos

• APPENDIX
• SUPPORT SLIDES

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

51
Real Options and Asymmetry

• At the expiration the option (F) only shall be
  exercised if V gt D
• The option creates an asymmetry, because the
  losses are limited to the cost to aquire the
  option and the upside is theoretically unlimited.
  The asymmetric effect is as greater as uncertain
  is the future value of the underlying asset V.
• At the expiration (T)

• For investment projects, V - D is the NPV and so
  it is possible to think the option value as F(t
  T) Max. (NPV, 0)

52
Example in EP with the Options Lens

• In a negotiation, important mistakes can be done
  if we dont 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

53
When Real Options Are Valuable?

• Flexibilities (real options) value greatest when
  we have
• High uncertainty about the future
• Very likely to receive relevant new information
  over time.
• Information can be costly (investment in
  information) or free .
• High room for managerial flexibility
• Allows management to respond appropriately to
  this new information
• Examples to expand or to contract the project
  better fitted development investment, etc.
• Projects with NPV around zero
• Flexibility to change course is more likely to be
  used and therefore is more valuable
• Under these conditions, the difference between
  real options analysis and other decision tools is
  substantial Tom Copeland

54
Geometric Brownian Motion Simulation

• The real simulation of a GBM uses the real drift
  a. The price P at future time (t 1), given the
  current value Pt is given by

• But for a derivative F(P) like the real option to
  develop an oilfiled, we need the risk-neutral
  simulation (assume the market is complete)
• The risk-neutral simulation of a GBM uses the
  risk-neutral drift a r - d . Why? Because by
  suppressing a risk-premium from the real drift a
  we get r - d. Proof
• Total return r r p (where p is the
  risk-premium, given by CAPM)
• But total return is also capital gain rate plus
  dividend yield r a d
• Hence, a d r p ? a - p r - d
  
• So, we use the risk-neutral equation below to
  simulate P

55
Other Parameters for the Simulation

• Other important parameters are the risk-free
  interest rate r and the dividend yield d (or
  convenience yield for commodities)
• Even more important is the difference r - d (the
  risk-neutral drift) or the relative value between
  r and d
• Pickles Smith (Energy Journal, 1993) suggest
  for long-run analysis (real options) to set r d
• We suggest that option valuations use,
  initially, the normal value of d, 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 d range from 4 to 8
  p.a.
• By using r d the risk-neutral drift is zero,
  which looks reasonable for a risk-neutral process

56
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 EX I
• The distribution of conditional expectations EX
  I is named here revelation distribution, that
  is, the distribution of RX EX 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) EX - g( I )2 . The choice of g that
  minimizes the error measure MSE(g) is exactly the
  conditional expectation EX I .
• This is a very known property used in
  econometrics
• The revelation distribution has nice practical
  properties (propositions)

57
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 EEX I ERX EX
  (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
• VarEX I VarRX VarX - EVarX 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

58
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)

• 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)

59
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

60
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
• E1BA1 100 100 (0.5 x 100) (0.5 x
  100) 300 MM bbl
• In case of negative revelation (50 chances) the
  expected value is
• E2BA1 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
• EA1RB 50 x E1(BA1) 50 x E2(BA1)
  250 million bbl EB
• As expected by Proposition 2
• And the variance of the revealed scenarios is
• VarA1RB 50 x (300 - 250)2 50 x (200 -
  250)2 2500 (MM bbl)2
• Let us check if the Proposition 3 was satisfied

61
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

62
Visualization of Revealed Scenarios Revelation
Distribution
All the revelation distributions have the same
mean (maringale) Prop. 4 OK!
63
Posterior Distribution x Revelation Distribution

• The picture below help us to answer the question
  Why learn?

64
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 gup
• Without full information about the size and
  productivity of the reserve, the non-optimized
  system doesnt permit to get the full project
  value

65
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
Eq B NPV q P B gup - D(B) if q B gt Eq
B NPV q P B gdown- D(B) if q B lt Eq B
In general we have gdown 1 and gup lt 1
66
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?

67
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, d, s change.
  But these factors dont 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

68
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
  alternative

The valuation is similar to the previously
presented Only by running the simulations is
possible to compare the higher NPV versus higher
flexibility
69
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.

70
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 Xs 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

71
Two Sequential Learning Schematic Tree

• Two sequential investment in information (wells
  B and C)

RevelationScenarios
PosteriorScenarios
InvestWell C
InvestWell B
NPV

400 300
350 (with 25 chances)
300
50
50

50
300 200
250 (with 50 chances)
100
50
50

50
200 100
- 200
150 (with 25 chances)

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

72
Visual FAQs 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
  wildcat
• Static NPV is negative and traditional theory
  recommends to give up the rights on the tract
• Real options will recommend to start the
  sequential investment, and depending of the
  information revealed, go ahead (exercise more
  options) or stop

73
Sequential Options (Dias, 1997)
Compact Decision-Tree
Note in million US
( Developed Reserves Value )
( Appraisal Investment 3 wells )
( Development Investment )
EMV - 15 20 x (400 - 50 - 300) ? EMV - 5
MM
( Wildcat Investment )

• 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.

74
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

75
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

2.25 stopping development 6 stopping
appraisal 8.25 total EMV gain
(Values in millions)
76
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(tT) max (NPV, 0) NPV V - D
77
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)

• 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

78
Brent Oil Prices Spot x Futures

• Note that the spot prices reach more extreme
  values than the long-term futures prices

79
Mean-Reversion Jumps for Oil Prices

• Adopted in the Marlim Project Finance (equity
  modeling) a mean-reverting process with jumps

(the probability of jumps)

• The jump size/direction are random f 2N
• In case of jump-up, prices are expected to
  double
• OBS E(f)up ln2 0.6931
• In case of jump-down, prices are expected to
  halve
• OBS ln(½) - ln2 - 0.6931

(jump size)
80
Equation for Mean-Reversion Jumps

• The interpretation of the jump-reversion equation
  is

discrete process (jumps)
continuous (diffusion) process
variation of the stochastic variable for time
interval dt
uncertainty from the continuous-time process
(reversion)


uncertainty from the discrete-time process (jumps)
mean-reversion drift positive drift if P lt
P negative drift if P gt P