Real Options and Investment under Uncertainty in PUC Rio

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Real Options and Investment under Uncertainty in PUC Rio Powered By Docstoc
					                  . Overview of
     Real Options in Petroleum
        Seminar Real Options in Real Life
 MIT/Sloan School of Management - May 2nd 2002

       By: Marco Antonio Guimarães Dias
             Petrobras and PUC-Rio, Brazil

Visit the first real options website: www.puc-rio.br/marco.ind/
                     Seminar Outline
u Introductionand overview of real options in
  upstream petroleum (exploration & production)
  l   Intuition, classical models, stochastic processes for oil prices
u Brazilian     applications of real options in petroleum
  l Timing of Petroleum Sector Policy (extendible options)
  l Petrobras research program called “PRAVAP-14”
    Valuation of Development Projects under Uncertainties
       Focus   on PUC-Rio projects.
u Investment      in information, real options and revelation
  l Combination of technical and market uncertainties
  l Assignment questions and the spreadsheet application
 Managerial View of Real Options (RO)
u RO is a modern methodology for economic evaluation
 of projects and investment decisions under uncertainty
  l   RO approach complements (not substitutes) the corporate tools (yet)
  l   Corporate diffusion of RO takes time and training
u RO  considers the uncertainties and the options (managerial
 flexibilities), giving two answers:
  l   The value of the investment opportunity (value of the option); and
  l   The optimal decision rule (threshold)
u RO    can be viewed as an optimization problem:
  l Maximize  the NPV (typical objective function) subject to:
  l (a) Market uncertainties (eg.: oil price);
  l (b) Technical uncertainties (eg., oil in place volume); and
  l (c) Relevant Options (managerial flexibilities)
Main Petroleum Real Options and Examples
            u   Option to Delay (Timing Option)
                l Wait,  see, learn, optimize before invest
                l Oilfield development; wildcat drilling

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

        u   Option to Expand the Production
            l   Depending of market scenario (oil prices, rig rates)
                and the petroleum reservoir behavior, new wells
                can be added to the production system
     E&P as a Sequential Real Options Process
Oil/Gas Success
Probability = p       Concession: Option to Drill the Wildcat
Expected Volume                    Exploratory (wildcat)
of Reserves = B
                                       Investment
       Revised
       Volume = B’
                      Undelineated Field: Option to Appraise
                                   Appraisal Investment


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


                      Developed Reserves: Options to Expand,
                       to Stop Temporally, and to Abandon.
Intuition (1): Timing Option and Oilfiled Value
u   Assume a simple equation for the oilfield development NPV:
    l   NPV = q B P - D = 0.2 x 500 x 18 – 1850 = - 50 million $
    l   Do you sell the oilfield for US$ 4 million?
    l   Suppose the following two-periods problem and only two scenarios
        in the second period for oil prices P.
                                             t=1
                                                P+ = 19  NPV = + 50 million $
                            50%


         t=0
    E[P] = 18 $/bbl               50%
NPV(t=0) = - 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 $
Intuition (2): Timing Option and Waiting Value
u Suppose  the same case but with a small positive NPV.
   What is better: develop now or wait and see?
    l   NPV = q B P - D = 0.2 x 500 x 18 – 1750 = + 50 million $
    l   Discount rate = 10%            t=1
                                                P+ = 19  NPV+ = + 150 million $
                            50%


         t=0
    E[P] = 18 /bbl                50%
NPV(t=0) = + 50 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
 Hence is better to wait and see, exercising the option only in favorable scenario
 Intuition (3): Deep-in-the-Money Real Option
u Suppose     the same case but with a higher NPV.
   l   What is better: develop now or wait and see?
   l   NPV = q B P - D = 0.25 x 500 x 18 – 1750 = + 500 million $
   l   Discount rate = 10%            t=1
                                               P+ = 19  NPV = 625 million $
                           50%


        t=0
    E[P] = 18 /bbl               50%
NPV(t=0) = 500 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
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?
u   Based on the textbook “Real Options” by Copeland & Antikarov
    l   Real options are as valuable as greater are the uncertainties and the flexibility to
        respond

                                               Low      Likelihood of receiving new information       High
         High                                                       Uncertainty
                               Managerial Flexibility



                                                           Moderate                    High
          Ability to respond




                                                        Flexibility Value         Flexibility Value
                                   Room for




                                                             Low                     Moderate
                                                        Flexibility Value         Flexibility Value

        Low
Classical Real Options in Petroleum Model
u Paddock & Siegel & Smith wrote a series of papers on
 valuation of offshore reserves in 80’s (published in 87/88)
  l   It is the best known model for oilfields development decisions
  l   It explores the analogy financial options with real options
  l   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 (d)

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

       Stock Volatility                      Volatility of Developed Reserve Value (s)

       Time to Expiration of the Option      Time to Expiration of the Investment Rights (t)
   Estimating the Model Parameters
u How   to estimate the value of underlying asset V?
 l 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”;
    Let us call q = economic quality of the developed reserve
        – The developed reserve value V is an increasing function of q
 l Discounted     cash flow estimate of V, that is:
    NPV = V - D  V = NPV + D
    It is possible to work with the entire cash-flows, but we can
    simplify this job identifying the main sources of value for V
    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
    Let us write the value V = q P B or NPV = q P B - D
NPV x P Chart and the Quality of Reserve
                      Linear Equation for the NPV:
    NPV (million $)
                            NPV = q P B - D


                                                               NPV in function of P




                                           tangent q = q . B

                                                               P ($/bbl)


   -D


                           The quality of reserve (q) is related
                           with the inclination of the NPV line
        Estimating the Model Parameters
u If   V = k P, we have sV = sP and dV = dP (D&P p.178. Why?)
   l   Risk-neutral Geometric Brownian: dV = (r - dV) V dt + sV V dz
u Volatility of        long-term oil prices (~ 20% p.a.)
   l   For development decisions the value of the benefit is linked to
       the long-term oil prices, not the (more volatile) spot prices
   l   A good market proxy is the longest maturity contract in futures
       markets with liquidity (Nymex 18th month; Brent 12th month)
   l   Volatily = standard-deviation of ( Ln Pt - Ln Pt-1 )
u Dividend        yield (or long-term convenience yield) ~ 6% p.a.
   l   Paddock & Siegel & Smith: equation using cash-flows
   l   If V = k P, we can estimate d from oil prices futures market
u Pickles     & Smith’s Rule (1993): r = d (in the long-run)
   l   “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
u   Note that the spot prices reach more extreme values and have more
    ‘nervous’ movements (more volatile) than the long-term futures prices
                                                                                      WTI Nymex Prices: Spot (First Month) vs. 18 Months
                                                                                                    Jul/1996 - Jan/2002
                          40

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

                          35                                  WTI Nymex Mth18 Close (US$/bbl)



                          30
          WTI (US$/bbl)




                          25



                          20



                          15



                          10



                          5
                                           10/22/1996




                                                                                            10/22/1997




                                                                                                                                             10/22/1998




                                                                                                                                                                                              10/22/1999




                                                                                                                                                                                                                                               10/22/2000




                                                                                                                                                                                                                                                                                                10/22/2001
                               7/22/1996



                                                        1/22/1997

                                                                    4/22/1997

                                                                                7/22/1997



                                                                                                         1/22/1998

                                                                                                                     4/22/1998

                                                                                                                                 7/22/1998



                                                                                                                                                          1/22/1999

                                                                                                                                                                      4/22/1999

                                                                                                                                                                                  7/22/1999



                                                                                                                                                                                                           1/22/2000

                                                                                                                                                                                                                       4/22/2000

                                                                                                                                                                                                                                   7/22/2000



                                                                                                                                                                                                                                                            1/22/2001

                                                                                                                                                                                                                                                                        4/22/2001

                                                                                                                                                                                                                                                                                    7/22/2001



                                                                                                                                                                                                                                                                                                             1/22/2002
Equation of the Undeveloped Reserve (F)
u Partial   (t, V) Differential Equation (PDE) for the option F

      0.5 s2 V2 FVV + (r - d) V FV - r F = - Ft

u Boundary     Conditions:
                                                   Managerial Action Is
                                                  Inserted into the Model
  l For V = 0, F (0, t) = 0
  l For t = T, F (V, T) = max [V - D, 0] = max [NPV, 0]
  l For V = V*, F (V*, t) = V* - D
  l “Smooth Pasting”, FV (V*, t) = 1     } Conditions at the Point of
                                           Optimal Early Investment

u Parameters: V =  value of developed reserve (eg., V = q P B);
  D = development cost; r = risk-free discount rate;
  d = dividend yield for V ; s = volatility of V
The Undeveloped Oilfield Value: Real Options and NPV
u Assume that V = q B P, so that we can use chart F x V or F x P
u Suppose the development break-even (NPV = 0) occurs at US$15/bbl
Threshold Curve: The Optimal Decision Rule
u Ator above the threshold line, is optimal the immediate
 development. Below the line: “wait, learn and see”
    Stochastic Processes for Oil Prices: GBM
u   Like Black-Scholes-Merton equation, the classic model of
    Paddock et al uses the popular Geometric Brownian Motion
    l   Prices have a log-normal distribution in every future time;
    l   Expected curve is a exponential growth (or decline);
    l   In this model the variance grows with the time horizon
               Mean-Reverting Process
u Inthis process, the price tends to revert towards a long-
  run average price (or an equilibrium level) P.
  l    Model analogy: spring (reversion force is proportional to the
       distance between current position and the equilibrium level).
  l    In this case, variance initially grows and stabilize afterwards
Stochastic Processes Alternatives for Oil Prices
u   There are many models of stochastic processes for oil prices
    in real options literature. I classify them into three classes.




u   The nice properties of Geometric Brownian Motion (few parameters,
    homogeneity) is a great incentive to use it in real options applications.
     l Pindyck (1999) wrote: “the GBM assumption is unlikely to lead to
       large errors in the optimal investment rule”
Mean-Reversion + Jump: the Sample Paths
u   100 sample paths for mean-reversion + jumps (l = 1 jump each 5 years)
Nominal Prices for Brent and Similar Oils (1970-2001)
u   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
u We(Dias & Rocha, 1998/9) adapt the Merton (1976)
 jump-diffusion idea for the oil prices case, considering:
  l   Normal news cause only marginal adjustment in oil prices,
      modeled with the continuous-time process of mean-reversion
  l   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)
uA 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)
  l   Win-win deal (higher oil prices  higher spread, and vice versa)
  l   Deal was in December 1998 when oil price was 10 US$/bbl
        The expected oil prices curve was a fast reversion towards US$ 20/bbl
        With the jumps possibility, we put a “collar” in the spread (cap and floor)
           – This jumps insight was very important because few months later the oil
             prices jump, doubling the value in Aug/99: the cap protected Petrobras
 Brazilian Timing Policy for the Oil Sector
u TheBrazilian petroleum sector opening started in 1997,
 breaking the Petrobras’ monopoly. For E&P case:
  l   Fiscal regime of concessions, with first-price sealed bid (like USA)
  l   Adopted the concept of extendible options (two or three periods).
        The  time extension is conditional to additional exploratory commitment
         (1-3 wells), established before the bid (it is not like Antamina)
  l The extendible feature occurred also in USA (5 + 3 years, for
    some areas of GoM) and in Europe (see paper of Kemna, 1993)
  l Options with extendible maturities was studied by Longstaff
    (1990) for financial applications
  l The timing for exploratory phase (time to expiration for the
    development rights) was object of a public debate
        TheNational Petroleum Agency posted the first project for debate in its
         website in February/1998, with 3 + 2 years, time we considered too short
        Dias   & Rocha wrote a paper on this subject, presented first in May 1998.
The Extendible Maturity Feature (2 Periods)
      Period        Available Options
           t = 0 to T1:
                           [Develop Now] or [Wait and See]
           First Period
T I M E




             T1: First    [Develop Now] or [Extend (commit K)]
            Expiration    or [Give-up (Return to Government)]

            T1 to T2:
                           [Develop Now] or [Wait and See]
          Second Period
           T2: Second      [Develop Now] or
           Expiration      [Give-up (Return to Government)]
Extendible Option Payoff at the First Expiration
u At the first expiration (T1), the firm can develop the field,
  or extend the option, or give-up/back to National Agency
u For geometric Brownian motion, the payoff at T1 is:
          The Options and Payoffs for Both Periods
             Using Mean-Reversion with Jumps
                                Options Charts
             Period
            t = 0 to T1:
            First Period

              T1: First
T I M E




             Expiration

             T1 to T2:
           Second Period

            T2: Second
            Expiration
Debate of Timing of Petroleum Policy
u   The oil companies considered very short the time of 3 + 2
    years that appeared in the first draft by National Agency
    l It was below the international practice mainly for deepwaters
      areas (e.g., USA/GoM: some areas 5 + 3 years; others 10 years)
    l During 1998 and part of 1999, the Director of the National
      Petroleum Agency (ANP) insisted in this short timing policy
    l The numerical simulations of our paper (Dias & Rocha, 1998)
      concludes that the optimal timing policy should be 8 to 10 years
    l In January 1999 we sent our paper to the notable economist,
      politic and ex-Minister Delfim Netto, highlighting this conclusion
    l In April/99 (3 months before the first bid), Delfim Netto wrote
      an article at Folha de São Paulo (a top Brazilian newspaper)
      defending a longer timing policy for petroleum sector
    l Delfim used our paper conclusions to support his view!
    l Few days after, the ANP Director finally changed his position!
           Since the 1st bid most areas have 9 years. At least it’s a coincidence!
  Alternatives Timing Policies in Dias & Rocha
u The table below presents the sensibility analysis for
 different timing policies for the petroleum sector




  l Option values (F) are proxy for bonus in the bid
  l Higher thresholds (P*) means more delay for investments
       Longer timing   means more bonus but more delay (tradeoff)
u Table
      indicates a higher % gain for option value
 (bonus) than a % increase in thresholds (delay)
  l   So, is reasonable to consider something between 8-10 years
PRAVAP-14: Some Real Options Projects
u PRAVAP-14      is a systemic research program named
    Valuation of Development Projects under Uncertainties
    l   I coordinate this systemic project by Petrobras/E&P-Corporative
u I’ll   present some real options projects developed:
    l   Selection of mutually exclusive alternatives of development
        investment under oil prices uncertainty (with PUC-Rio)
    l   Exploratory revelation with focus in bids (pre-PRAVAP-14)
    l   Dynamic value of information for development projects
    l   Analysis of alternatives of development with option to expand,
        considering both oil price and technical uncertainties (with PUC)
u   We analyze different stochastic processes and solution methods
    l   Geometric Brownian, reversion + jumps, different mean-reversion models
    l   Finite differences, Monte Carlo for American options, genetic algorithms
    l   Genetic algorithms are used for optimization (thresholds curves
        evolution)
Oil/Gas Success
                         E&P Process and Options
Probability = p      u   Drill the wildcat (pioneer)? Wait and See?
Expected Volume
of Reserves = B
                     u   Revelation: additional waiting incentives
       Revised
       Volume = B’   u   Appraisal phase: delineation of reserves
                     u   Invest in additional information?

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

                     u Developed Reserves.
                     u Expand the production?
                       Stop Temporally? Abandon?
Selection of Alternatives under Uncertainty
u 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.
   l   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
   l   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
u Hence, an important investment decision is:
  l How select the best one from a set of mutually exclusive alternatives?
    Or, What is the best intensity of investment for a specific oilfield?
  l I follow the paper of Dixit (1993), but considering finite-lived options.
The Best Alternative at Expiration (Now or Never)
u   The chart below presents the “now-or-never” case for three
    alternatives. In this case, the NPV rule holds (choose the higher one).
     l   Alternatives: A1(D1, q1); A2(D1, q1); A3(D3, q3), with D1 < D2 < D3 and q1 < q2 < q3




u   Hence, the best alternative depends on the oil price P. However, P is uncertain!
The Best Alternative Before the Expiration
u   Imagine that we have t years before the expiration and in
    addition the long-run oil prices follow the geometric Brownian
    l   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.
                                      u   The decision rule is:
                                          l   If P < P*2 , “wait and see”
                                                  Alone, A1 can be even deep-in-the-money,
                                                   but wait for A2 is more valuable
                                          l   If P = P*2 , invest now with A2
                                                Wait is   not more valuable
                                          l   If P > P*2 , invest now with the higher
                                              NPV alternative (A2 or A3 )
                                                Depending of     P, exercise A2 or A3
                                      u   How about the decision rule along
                                          the time? (thresholds curve)
                                          l   Let us see from a PRAVAP-14 software
  Threshold Curves for Three Alternatives
u There
      are regions of wait and see and others that the
 immediate investment is optimal for each alternative




      Investments
      D3 > D2 > D1
     Technical Uncertainty: A Dynamic View
u Before see the others applications, is necessary to discuss the
  technical uncertainties with the dynamic real options lens
u 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?
    l   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
u How     to model the technical uncertainty and its evolution
    after one or more investment in information?
    l   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
  Technical Uncertainty and Risk Reduction
u Technical uncertainty decreases when efficient investments
  in information are performed (learning process).
u Suppose a new basin with large geological uncertainty. It is
  reduced by the exploratory investment of the whole industry
   l   The “cone of uncertainty” (Amram & Kulatilaka) can be adapted to
       understand the technical uncertainty:
         Higher
         Risk
                                                                   Lower
                                                                   Risk
             confidence
               interval




 Expected                                                               Expected
 Value                    Lack of Knowledge Trunk of Cone               Value
                                                                  Project
                                 Risk reduction by the            evaluation
       Current                   investment in information        with additional
       project                   of all firms in the basin        information
       evaluation                (driver is the investment, not   (t = T)
       (t=0)                     the passage of time directly)
        Technical Uncertainty and Revelation
u   But in addition to the risk reduction process, there is another
    important issue: revision of expectations (revelation process)
    l   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
                                                                      t=T
                                                                 Value with
                                                                 good revelation



E[V]                                                             Value with
                                                                 neutral revelation


                                                                  Value with
                                                                  bad revelation

 Current project            Investment in               Project value
 evaluation (t=0)           Information                 after investment
Oil/Gas Success
                         E&P Process and Options
Probability = p      u Drill the wildcat (pioneer)? Wait and See?
Expected Volume
of Reserves = B      u Revelation: additional waiting incentives

       Revised
       Volume = B’   u Appraisal phase: delineation of reserves
                     u Invest in additional information?


                     u Delineated but Undeveloped Reserves.
                     u Develop? “Wait and See” for better
                       conditions? What is the best alternative?
                     u Developed Reserves.
                     u Expand the production?
                       Stop Temporally? Abandon?
      Technical Uncertainty in New Basins
u   The number of possible scenarios to be revealed (new expectations)
    is proportional to the cumulative investment in information
     l Information can be costly (our investment) or free, from the other
        firms investment (free-rider) in this under-explored basin

           Investment                   Investment in information       Revelation
           in information               (costly and free-rider)         Distribution
           (wildcat drilling, etc.)
                                                                         Possible
                                                                         scenarios

      .
     t=0               t=1                                        t=T
                                                                         after the
                                                                         information
                                                                         arrived
    Today
                                                                         during the
    technical
                                      Possible scenarios                 option lease
    and economic
                                      after the information              term
    valuation
                                      arrived during the
                                      first year of option term
u    The arrival of information process leverage the option value of a tract
        Valuation of Exploratory Prospect
u Suppose        that the firm has 5 years option to drill the wildcat
    l   Other firm wants to buy the rights of the tract for $ 3 million $.
          Do   you sell? How valuable is the prospect?

   “Compact Tree”
                                         E[B] = 150 million barrels (expected reserve size)


                             Success
                                         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
                      Dry Hole
20 million $
(IW = wildcat                          EMV = Expected Monetary Value = - IW + (CF . NPV) 
 investment)                            EMV = - 20 + (15% . 100) = - 5 million $

                                        Do you sell the prospect rights for US$ 3 million?
Monte Carlo Combination of Uncertainties
u   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:
    l   The expectations in 5 years almost surely will change and so the prospect value
    l   The revelation distributions and the risk-neutral distribution for oil prices are:
             Distribution of Expectations
              (Revelation Distributions)
        A Visual Equation for Real Options
u 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.




                                                +
                                                             Prospect Evaluation
                                                                    (in million $)
                                                            Traditional Value = - 5


  =                                                      Options Value (at T) = + 12.5
                                                         Options Value (at t=0) = + 7.6

                                                         So, refuse the $ 3 million offer!
Oil/Gas Success
                         E&P Process and Options
Probability = p      u   Drill the wildcat (pioneer)? Wait and See?
Expected Volume
of Reserves = B
                     u   Revelation: additional waiting incentives
       Revised
       Volume = B’   u   Appraisal phase: delineation of reserves
                     u   Invest in additional information?

                     u Delineated but Undeveloped Reserves.
                     u Develop? “Wait and See” for better
                       conditions? What is the best alternative?
                     u Developed Reserves.
                     u Expand the production?
                       Stop Temporally? Abandon?
 Relevance of the Revelation Distribution
u Investments  in information permit both a reduction of the
  uncertainty and a revision of our expectations on the basic
  technical parameters. Let us answer assignment question 1.b
  l   Firms use the new expectation to calculate the NPV or the real options
      exercise payoff. This new expectation is conditional to information.
  l   When we are evaluating the investment in information, the conditional
      expectation of the parameter X is itself a random variable E[X | I]
  l   The distribution of conditional expectations E[X | I] is named here
      revelation distribution, that is, the distribution of RX = E[X | I]
  l   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
  l   The revelation distribution has nice practical properties (propositions)
     The Revelation Distribution Properties
u   Full revelation definition: when new information reveal all the
    truth about the technical parameter, we have full revelation
    l   Much more common is the partial revelation case, but full revelation is
        important as the limit goal for any investment in information process
u   The revelation distributions RX (or distributions of conditional
    expectations with the new information) have at least 4 nice
    properties for the real options practitioner:
    l   Proposition 1: for the full revelation case, the distribution of revelation RX is
        equal to the unconditional (prior) distribution of X
    l   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)
    l   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
    l   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
u Suppose   the following stylized case of investment in
    information in order to get intuition on the propositions
    l   Only one well was drilled, proving 100 MM bbl (MM = million)

                                                             Area B: possible
Area A: proved               A                B              50% chances of
BA = 100 MM bbl
                                                            BB = 100 MM bbl
                                                            & 50% of nothing

 Area C: possible           C                                Area D: possible
 50% chances of
                                              D              50% chances of
BC = 100 MM bbl                                             BD = 100 MM bbl
& 50% of nothing                                            & 50% of nothing


u   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
u Alternative    Zero: Not invest in information
  l   This case there is only a single scenario, the current expectation
  l   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
u 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.
  l   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
u The   variance of this prior distribution is 7500 (million bbl)2
Alternative 1: Invest in Information with Only One Well
u Suppose that we drill only the well in the area B.
    l   This case generated 2 scenarios, because the well B result can be
        either dry (50% chances) or success proving more 100 MM bbl
    l   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
    l   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
    l   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)
u   So, the expected value of the revelation distribution is:
    l   EA1[RB] = 50% x E1(B|A1) + 50% x E2(B|A1) = 250 million bbl = E[B]
          As   expected by Proposition 2
u   And the variance of the revealed scenarios is:
    l   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
u In order to check the Proposition 3, we need to calculated
  the expected reduction of variance with the alternative A1
u The prior variance was calculated before (7500).
u The posterior variance has two cases for the well B outcome:
    l   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)
    l   The negative revelation case is analog: can occur 100 MM bbl (25%
        chances); 200 MM bbl (50%); and 300 MM bbl (25%)
    l   The residual variance in both scenarios are 5000 (MM bbl)2
    l   So, the expected variance of posterior distribution is also 5000
u   So, the expected reduction of uncertainty with the alternative
    A1 is: 7500 – 5000 = 2500 (MM bbl)2
    l   Equal variance of revelation distribution(!), as expected by Proposition 3
Visualization of Revealed Scenarios: Revelation Distribution




                                                                 All the revelation distributions have the same mean (maringale): Prop. 4 OK!
     This is exactly the prior distribution of B (Prop. 1 OK!)
Posterior Distribution x Revelation Distribution
u   The picture below help us to answer the assignment question 1.a
                                                        Why learn?

                                                        Reduction
                                                        of technical
                                                        uncertainty


                                                            
                                                        Increase the
                                                        variance of
                                                        revelation
                                                        distribution
                                                        (and so the
                                                        option value)
Revelation Distribution and the Experts
u   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:
    l   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.
    l   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
u   In addition, the discounted cash flow analyst together with the
    reservoir engineer, need to find the penalty factor gup:
    l   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
u   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.
    l   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 gup - D(B) if q B > E[q B]
                                       NPV = q P B gdown- D(B) if q B < E[q B]

                                        In general we have gdown = 1 and gup < 1
    Geometric Brownian Motion Simulation
u   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:
     Pt+1 = Pt exp{ (a - 0.5 s2) Dt + s N(0, 1)                       Dt   }
    l   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)
u   The risk-neutral simulation of a GBM uses the risk-neutral
    drift a’ = r - d . Why? Because by supressing a risk-premium
    from the real drift a we get r - d. Proof:
    l   Total return r = r + p (where p is the risk-premium, given by CAPM)
    l   But total return is also capital gain rate plus dividend yield: r = a + d
    l   Hence, a + d = r + p  a - p = r - d
u   So, we use the risk-neutral equation below to simulate P
    Pt+1 = Pt exp{ (r - d - 0.5 s2) Dt + s N(0, 1)                         Dt   }
                             Real x Risk-Neutral Simulation
u   The GBM simulation paths: one real (a) and the other risk-
    neutral (r - d). Note that the risk-neutral is below the real one.
                                                      Real Versus Risk-Neutral Simulations
                        45


                        40                                 Real Simulation
                                                           Risk-Neutral Simulation
                        35


                        30
    Oil Price ($/bbl)




                        25


                        20


                        15


                        10


                        5


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                                                                                      Time (Years)
  Oil Price Process x Revelation Process
u Let   us answer the assignment question 1.c
  l   Oil price (and other market uncertainties) evolves continually along
      the time and it is non-controllable by oil companies (non-OPEC)
  l   Revelation distributions occur as result of events (investment in
      information) in discrete points along the time
         For exploration of new basins sometimes the revelation of information from
          other firms can be relevant (free-rider), but it also occurs in discrete-time
         In many cases (appraisal phase) only our investment in information is
          relevant and it is totally controllable by us (activated by management)
  l   In short, every day the oil prices changes, but our expectation about
      the reserve size will change only when an investment in information
      is performed  so the expectation can remain the same for months!

  P
                                                 Inv
 E[B]
                                 Inv
The Normalized Threshold and Valuation
u Assignment question 1.d is about valuation under optimization
u 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”
u Assume D as a function of B but approximately independent
  of q. Assume the linear equation: D = 310 + (2.1 x B) (MM$)
u This means that if B varies, the exercise price D of our option
  also varies, and so the threshold V*.
    l   The computational time for V* is much higher than for D
u   We need perform a Monte Carlo simulation to combine the
    uncertainties after an information revelation.
    l   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
u   In order to reduce the computational time, we work with the
    normalized threshold (V/D)*. Why?
     Normalized Threshold and Valuation
u   We will perform the valuation considering the optimal
    exercise at the normalized threshold level (V/D)*
    l   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)*
u   Advantage: (V/D)* is homogeneous of degree 0 in V and D.
    l   This means that the rule (V/D)* remains valid for any V and D
    l   So, for any revealed scenario of B, changing D, the rule (V/D)* remains
    l   This was proved only for geometric Brownian motions
    l   (V/D)*(t) changes only if the risk-neutral stochastic process parameters
        r, d, s change. But these factors don’t change at Monte Carlo simulation
u   The computational time of using (V/D)* is much lower than V*
    l   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
     Combination of Uncertainties in Real Options
u   The simulated sample paths are checked with the threshold V/D*


                                                    A




                                                                             B




                     Present Value (t = 0)                             F(t = 2) = 0
F(t = 0) =                                   Option F(t = 1) = V - D
= F(t=1) * exp (- r*t)                                                   Expired
                                                                         Worthless
              Overall x Phased Development
u   Assignment question 1.e is about two alternatives
    l   Overall development has higher NPV due to the gain of scale
    l   Phased development has higher capacity to use the information along
        the time, but lower NPV
u   With the information revelation from Phase 1, we can
    optimize the project for the Phase 2
    l   In addition, depending of the oil price scenario and other market and
        technical conditions, we can not exercise the Phase 2 option
    l   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
           Spreadsheet Application
u Assignment   Part 2
u Let us see the spreadsheet timing_inv_inf-hqr-MIT.xls
u It permits to choose the best alternative of investment in
  information (and check if is better to invest in information
  or not)
u It calculates the dynamic net value of information
Oil/Gas Success
                         E&P Process and Options
Probability = p      u Drill   the wildcat? Wait? Extend?
Expected Volume
                     u   Revelation, option-game: waiting incentives
of Reserves = B

       Revised
       Volume = B’   u Appraisal phase: delineation of reserves
                     u Technical uncertainty: sequential options

                     u Delineated but Undeveloped Reserves.
                     u Develop? What the Best Alternative?
                       Wait and See? Extend the option?

                      u Developed Reserves.
                      u Expand the production?
                      u Stop Temporally? Abandon?
       Option to Expand the Production
u Analyzing a large ultra-deepwater project in Campos
  Basin, Brazil, we faced two problems:
  l   Remaining technical uncertainty of reservoirs is still important.
        Inthis 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.
  l   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.
u Solution:     leave these wells as optional wells
  l   Buy flexibility with an additional investment in the production
      system: platform with capacity to expand (free area and load)
  l   It permits a fast and low cost future integration of these wells
        Theexercise 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
u 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




u Thepractical “now-or-never” is mainly because in many
 cases the effect of secondary depletion is relevant
  l   The oil migrates from the original area so that the exercise of the
      option gradually become less probable (decreasing NPV)
        Inaddition, distant exercise of the option has small present value
        Recall the expenses to embed flexibility occur between t = 0 and t = 3
Secondary Depletion Effect: A Complication
u   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)




              petroleum reservoir (top view) and the grid of wells
u   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
u Define the quantity of wells “deep-in-the-money” to start
  the basic investment in development
u Define the maximum number of optional wells
u Define the timing (accumulated production) that reservoir
  information will be revealed and the revelation distributions
u Define for each revealed scenario the marginal production
  of each optional well as function of time.
   l   Consider the secondary depletion if we wait after learn about reservoir
u Add  market uncertainty (stochastic process for oil prices)
u Combine uncertainties using Monte Carlo simulation
u Use an optimization method to consider the earlier exercise
  of the option to drill the wells, and calculate option value
   l   Monte Carlo for American options is a growing research area
   l   Many Petrobras-PUC projects use Monte Carlo for American options
                          Conclusions
u The real options models in petroleum bring a rich
  framework to consider optimal investment under
  uncertainty, recognizing the managerial flexibilities
  l    Traditional discounted cash flow is very limited and can
      induce to serious errors in negotiations and decisions
u We  saw the classical model, working with the intuition and
  the real options toolkit
  l   We saw different stochastic processes and other models
uI gave an idea about the real options research at Petrobras
  and PUC-Rio
u We worked more in models of value of information
  combining technical uncertainties with market uncertainty
  l   The model using the revelation distribution gives the correct
      incentives for investment in information
u Thank    you very much for your time
                              Anexos

          APPENDIX
    SUPPORT SLIDES
u   See more on real options in the first website on real options at:
               http://www.puc-rio.br/marco.ind/
Comparing Jump-Reversion with GBM
u   Jump-reversion points lower thresholds for longer maturity
u   The threshold discontinuity near of T2 is due the behavior of d, that
    can be negative for lower values of P: d = r - h( P - P)
    l   A necessary condition for early exercise of American option is d > 0
Oil Drilling Bayesian Game (Dias, 1997)
u   Oil exploration: with two or few oil companies exploring a
    basin, can be important to consider the waiting game of drilling
u   Two companies X and Y with neighbor tracts and correlated oil
    prospects: drilling reveal information
    l   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.
    l   In this case the effect of the competitor presence is to increase the
        value of waiting to invest

               Company X tract                Company Y tract
    Two Sequential Learning: Schematic Tree
u   Two sequential investment in information (wells “B” and “C”):

     Invest         Invest              Posterior      Revelation            NPV
     Well “B”       Well “C”            Scenarios      Scenarios

                                         {   400
                                             300
                                                    350 (with 25% chances)    300




                                         {   300
                                             200
                                                    250 (with 50% chances)    100




                                         {   200
                                             100
                                                    150 (with 25% chances)   - 200


u   The upper branch means good news, whereas the lower one means bad news
       Visual FAQ’s on Real Options: 9
u Ispossible real options theory to recommend
  investment in a negative NPV project?

Answer:  yes, mainly sequential options with
  investment revealing new informations
  l    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
         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 )
u   Traditional method, looking only expected values, undervaluate
    the prospect (EMV = - 5 MM US$):
    l   There are sequential options, not sequential obligations;
    l   There are uncertainties, not a single scenario.
Sequential Options and Uncertainty
                  u   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
u   Assume it is a “now or
    never” option
u If we get continuous bad
  news, is better to stop
  investment
u Sequential options turns
  the EMV to a positive
  value
u   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)
   Economic Quality of the Developed Reserve
u Imagine   that you want to buy 100 million barrels of developed
  oil reserves. Suppose a long run oil price is 20 US$/bbl.
   l   How much you shall pay for the barrel of developed reserve?
u One  reserve in the same country, water depth, oil quality,
  OPEX, etc., is more valuable than other if is possible to extract
  faster (higher productivity index, higher quantity of wells)
uA  reserve located in a country with lower fiscal charge and
  lower risk, is more valuable (eg., USA x Angola)
u 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
   l Where q = economic quality of the developed reserve
   l The value of the developed reserve is v times the reserve size (B)
Monte Carlo Simulation of Uncertainties
u Simulation will combine uncertainties (technical and market) for
  the equation of option exercise: NPV(t)dyn = q . B . P(t) - D(B)

         Parameter                Distribution         Values (example)
  Economic Quality of the                            Minimum = 10%
  Developed Reserve (q)                              Most Likely = 15%
  (only at t = trevelation)                          Maximum = 20%
  Reserve Size (B) (only                              Minimum = 300
  at t = trevelation)                                 Most Likely = 500
  (in million of barrels)                             Maximum = 700
                                                     Mean = 18 US$/bbl
   Oil Price (P) ($/bbl)                             Standard-Deviation:
  (from t = 0 until t = T)                           changes with the time
  l 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
Real Options Evaluation by Simulation + Threshold Curve
u   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




                                                                                               B




      F(t = 0) =            Present Value (t = 0)   Option F(t = 5.5) = V - D      F(t = 8) = 0
      = F(t=5.5) * exp (- r*t)                                                  Expires Worthless
 Mean-Reversion + Jumps for Oil Prices
u Adopted in the Marlim Project Finance (equity
  modeling) a mean-reverting process with jumps:


         where:
                                       (the probability of jumps)


u The  jump size/direction
  are random: f ~ 2N
l In case of jump-up, prices
  are expected to double
      OBS: E(f)up = ln2 = 0.6931
l In case of jump-down, prices
  are expected to halve
      OBS: ln(½) = - ln2 = - 0.6931                  (jump size)
Equation for Mean-Reversion + Jumps
u The    interpretation of the jump-reversion equation is:
                                                                              discrete
                                                                              process
                                  continuous (diffusion) process              (jumps)

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




                                                                   uncertainty from
                       mean-reversion drift:                       the discrete-time
                       positive drift if P < P                     process (jumps)
                       negative drift if P > P
         Brent Oil Prices: Spot x Futures
u   Note that the spot prices reach more extreme values than the
    long-term futures prices
                                                                                                     Brent Prices: Spot (Dated) vs. IPE 12 Month
                                                                                                                Jul/1996 - Jan/2002
                         40

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

                         35                                        Brent IPE Mth12 Close (US$/bbl)



                         30
       Brent (US$/bbl)




                         25



                         20



                         15



                         10



                         5
                                          10/22/1996




                                                                                            10/22/1997




                                                                                                                                             10/22/1998




                                                                                                                                                                                              10/22/1999




                                                                                                                                                                                                                                               10/22/2000




                                                                                                                                                                                                                                                                                                10/22/2001
                              7/22/1996



                                                       1/22/1997

                                                                    4/22/1997

                                                                                7/22/1997



                                                                                                         1/22/1998

                                                                                                                     4/22/1998

                                                                                                                                 7/22/1998



                                                                                                                                                          1/22/1999

                                                                                                                                                                      4/22/1999

                                                                                                                                                                                  7/22/1999



                                                                                                                                                                                                           1/22/2000

                                                                                                                                                                                                                       4/22/2000

                                                                                                                                                                                                                                   7/22/2000



                                                                                                                                                                                                                                                            1/22/2001

                                                                                                                                                                                                                                                                        4/22/2001

                                                                                                                                                                                                                                                                                    7/22/2001



                                                                                                                                                                                                                                                                                                             1/22/2002
Brent Oil Prices Volatility: Spot x Futures
u   Note that the spot prices volatility is much higher than the long-
    term futures volatility
                                                                                      Brent Volatility: Spot (Dated) vs. 12 Month (500 last data)
                                                                                                         Jul/1996 - Jan/2002
                               50%

                                                                   Brent Spot (Dated)
                               45%
                                                                   Brent IPE 12 Month

                               40%
         Volatility (% p.a.)




                               35%


                               30%


                               25%


                               20%


                               15%


                               10%
                                                 10/18/1996




                                                                                                    10/18/1997




                                                                                                                                                     10/18/1998




                                                                                                                                                                                                      10/18/1999




                                                                                                                                                                                                                                                       10/18/2000




                                                                                                                                                                                                                                                                                                        10/18/2001
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     Other Parameters for the Simulation
u Other    important parameters are the risk-free interest
    rate r and the dividend yield d (or convenience yield for
    commodities)
    l   Even more important is the difference r - d (the risk-neutral
        drift) or the relative value between r and d
u Pickles  & Smith (Energy Journal, 1993) suggest for
    long-run analysis (real options) to set r = d
    l   “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”
u Reasonable values for r and d range from 4 to 8% p.a.
u By using r = d the risk-neutral drift is zero, which looks
  reasonable for a risk-neutral process
Example in E&P with the Options Lens
u Ina negotiation, important mistakes can be done if we
  don´t consider the relevant options
u Consider two marginal oilfields, with 100 million bbl, both
  non-developed and both with NPV = - 3 millions in the
  current market conditions
  l    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
u Cia X offers US 1 million for the rights of each oilfield.
  Do you accept the offer?
u With the static NPV, these fields have no value and even
  worse, we cannot see differences between these two fields
  l    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
  l    In addition, the field B is more valuable (higher option) than the field A

				
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