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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 ($/bbl) 25 20 15 10 5 0 0 3 5 8 0 3 5 8 0 3 5 8 0 3 5 8 0 3 5 8 0 3 5 8 0 0. 0. 0. 0. 1. 1. 1. 1. 2. 2. 2. 2. 3. 3. 3. 3. 4. 4. 4. 4. 5. 5. 5. 5. 6. Time (Years) Oil Price Process x Revelation Process u Let us answer the assignment question 1.c l Oil price (and other market uncertainties) evolves continually along the time and it is non-controllable by oil companies (non-OPEC) l Revelation distributions occur as result of events (investment in information) in discrete points along the time For exploration of new basins sometimes the revelation of information from other firms can be relevant (free-rider), but it also occurs in discrete-time In many cases (appraisal phase) only our investment in information is relevant and it is totally controllable by us (activated by management) l In short, every day the oil prices changes, but our expectation about the reserve size will change only when an investment in information is performed so the expectation can remain the same for months! P Inv E[B] Inv The Normalized Threshold and Valuation u Assignment question 1.d is about valuation under optimization u Recall that the development option is optimally exercised at the threshold V*, when V is suficiently higher than D Exercise the option only if the project is “deep-in-the-money” u Assume D as a function of B but approximately independent of q. Assume the linear equation: D = 310 + (2.1 x B) (MM$) 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 7/18/1996 1/18/1997 4/18/1997 7/18/1997 1/18/1998 4/18/1998 7/18/1998 1/18/1999 4/18/1999 7/18/1999 1/18/2000 4/18/2000 7/18/2000 1/18/2001 4/18/2001 7/18/2001 1/18/2002 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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