. Real Options in Petroleum: An Overview Seminar Real Options in Real Life

By: Marco Antonio Guimarães Dias Petrobras and PUC-Rio, BrazilVisit the first real options website:www.puc-rio.br/marco.ind/ . Real Options in Petroleum:An Overview Seminar Real Options in Real Life MIT/Sloan School of Management - May 5th 2003

Seminar Outline • Introduction and overview of real options in upstream petroleum (exploration & production) • Intuition, classical models, stochastic processes for oil prices • Brazilian applications of real options in petroleum • Timing of Petroleum Sector Policy (extendible options) • Petrobras research program called “PRAVAP-14” Valuation of Development Projects under Uncertainties • Focus on PUC-Rio projects. • Investment in information, real options and revelation • Combination of technical and market uncertainties • Assignment questions and the spreadsheet application

Managerial View of Real Options (RO) • RO is a modern methodology for economic evaluation of projects and investment decisions under uncertainty • RO approach complements (not substitutes) the corporate tools (yet) • Corporate diffusion of RO takes time and training • RO considers the uncertainties and the options (managerial flexibilities), giving two answers: • The value of the investment opportunity (value of the option);and • The optimal decision rule (threshold) • RO can be viewed as an optimization problem: • Maximize the NPV(typical objective function) subject to: • (a) Market uncertainties (eg.: oil price); • (b) Technical uncertainties (eg., oil in place volume); and • (c) Relevant Options (managerial flexibilities)

When Real Options Are Valuable? Likelihood of receiving new information Low High U n c e r t a i n t y High High Flexibility Value Moderate Flexibility Value Room for Managerial Flexibility Ability to respond Moderate Flexibility Value Low Flexibility Value Low • Based on the textbook “Real Options” by Copeland & Antikarov • Real options are as valuable as greater are the uncertainties and the flexibility to respond

Main Petroleum Real Options and Examples • Option to Delay (Timing Option) • Wait, see, learn, optimize before invest • Oilfield development; wildcat drilling • Abandonment Option • Managers are not obligated to continue a business plan if it becomes unprofitable • Sequential appraisal program can be abandoned earlier if information generated is not favorable • Option to Expand the Production • Depending of market scenario (oil prices, rig rates) and the petroleum reservoir behavior, new wells can be added to the production system

E&P as a Sequential Real Options Process Oil/Gas Success Probability = p • Concession: Option to Drill the Wildcat Development Investment Appraisal Investment Exploratory (wildcat) Investment Expected Volume of Reserves = B Revised Volume = B’ • Undelineated Field: Option to Appraise • Delineated Undeveloped Reserves: Options to • Invest in Additional Information and to Develop • Developed Reserves: Options to Expand, to Stop Temporally, and to Abandon.

A Simple Equation for the Development NPV • Let us use a simple equation for the net present value (NPV) in our numerical examples. We can write NPV = V – D, where: • V = value of the developed reserve (PV of revenues net of OPEX & taxes) • D = development investment (also in PV, is the exercise price of the option) • Given a long-run expectation on oil-prices (P), how much we shall pay per barrel of developed reserve? • The value of one barrel of reserve depends of many variables (permo-porosity quality, oil/gas quality, discount rate, reserve location, etc.) • The relation between the value for one barrel of (sub-surface) developed reserve v and the (surface) oil price per barrel P is named the economic quality of the reserveq (because higher q means higher reserve value v) • Value of one barrel of developed reserve = v = q . P • Where q = economic quality of the developed reserve • The developed reserve value V is v times the reserve volume (B) • So, let us use the equation: NPV = V - D = q P B - D

NPV x P Chart and the Quality of Reserve Linear Equation for the NPV (“Business Model”): NPV = q P B - D NPV in function of P NPV (million $) spreadsheet value tangent q = q . B P ($/bbl) - D The quality of reserve (q) is relatedwith the NPV line inclination.

Intuition (1): Timing Option and Oilfiled Value t = 1 50% t = 0 50% • Suppose the following two-period problem and only two scenarios in the second period for oil prices P. Assume that simple equation for the oilfield development NPV: • NPV = q B P - D = 0.2 x 500 x 18 – 1850 = - 50 million $ • Do you sell the oilfield for US$ 4 million? E[P+] = 19  NPV+ = + 50 million $ E[P] = 18 /bbl NPV(t=0) = - 50 million $ E[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 t = 1 50% t = 0 E[P-] = 17  NPV- = - 50 million $ 50% Rational manager will not exercise this option  Max (NPV-, 0) = zero • Assume that simple equation for the oilfield development NPV • NPV = q B P - D = 0.2 x 500 x 18 – 1750 = + 50 million $ • What is the best decision: develop now or “wait and see”? • Discount rate = 10% E[P+] = 19  NPV+ = + 150 million $ E[P] = 18 /bbl NPV(t=0) = + 50 million $ 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 t = 1 50% t = 0 E[P-] = 17  NPV- = 120 million $ 50% • Suppose the same case but with a higher NPV. • NPV = q B P - D = 0.22 x 500 x 18 – 1750 = + 230 million $ • What is better: develop now or “wait and see”? • Discount rate = 10% E[P+] = 19  NPV+ = 340 million $ E[P] = 18 /bbl NPV(t=0) = 230 million $ Hence, at t = 1, the project NPV is: (50% x 340) + (50% x 120) = 230 million $ The present value is: NPVwait(t=0) = 230/1.1 = 209.1 < 230 Immediate exercise is optimal because this project is deep-in-the-money (high NPV) There is a NPV between 50 and 230 that value of wait = exercise now (threshold)

Classical Real Options in Petroleum Model 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) • Paddock & Siegel & Smith wrote a series of papers on valuation of offshore reserves in 80’s (published in 87/88) • It is the best known model for development decisions • It explores the analogy financial options with real options • Uncertainty is modeled using the Geometric Brownian Motion Time to Expiration of the Option Time to Expiration of the Investment Rights (t)

Equation of the Undeveloped Reserve (F) 0.5 s2 V2 FVV + (r - d) V FV- r F = - Ft Managerial Action Is Inserted into the Model } Conditions at the Point of Optimal Early Investment • For V = V*, F (V*, t) = V* - D • “Smooth Pasting”, FV (V*, t) = 1 • Parameters: V = value of developed reserve (eg., V = q P B); D = development cost; r = risk-free discount rate;d = dividend yield for V ; s = volatility of V • Partial (t, V) Differential Equation (PDE) for the option F • Boundary Conditions: • For V = 0, F (0, t) = 0 • For t = T, F (V, T) = max [V - D, 0] = max [NPV, 0]

The Undeveloped Oilfield Value: Real Options and NPV • Assume that V = q B P, so that we can use chart F x V or F x P • Suppose the development break-even (NPV = 0) occurs at US$15/bbl

Threshold Curve: The Optimal Decision Rule • At or above the threshold line, is optimal the immediate development. Below the line: “wait, learn and see”

Estimating the Model Parameters • If V = k P, we have sV = sP and dV = dP (D&P p.178. Why?) • Risk-neutral Geometric Brownian: dV = (r - dV) V dt + sV V dz • Volatility of long-term oil prices (~ 20% p.a.) • For development decisions the value of the benefit is linked to the long-term oil prices, not the (more volatile) spot prices • A good market proxy is the longest maturity contract in futures markets with liquidity (Nymex 18th month; Brent 12th month) • Volatily = standard-deviation of ( Ln Pt- Ln Pt-1 ) • Dividend yield (or long-term convenience yield) ~ 6% p.a. • Paddock & Siegel & Smith: equation using cash-flows • If V = k P, we can estimate d from oil prices futures market • Pickles & Smith’s Rule (1993): r = d (in the long-run) • “We suggest that option valuations use, initially, the ‘normal’ value of net convenience yield, which seems to equal approximately the risk-free nominal interest rate”

NYMEX-WTI Oil Prices: Spot x Futures • Note that the spot prices reach more extreme values and have more ‘nervous’ movements (more volatile) than the long-term futures prices Term Structure from Oil Futures Market

Stochastic Processes for Oil Prices: GBM • Like Black-Scholes-Merton equation, the classic model of Paddock et al uses the popular Geometric Brownian Motion • Prices have a log-normal distribution in every future time; • Expected curve is a exponential growth (or decline); • In this model the variance grows with the time horizon

Mean-Reverting Process • In this process, the price tends to revert towards a long-run average price (or an equilibrium level) P. • Model analogy: spring (reversion force is proportional to the distance between current position and the equilibrium level). • In this case, variance initially grows and stabilize afterwards

Stochastic Processes Alternatives for Oil Prices • There are many stochastic processes models for oil prices in the real options literature. I classify them into three classes. • The nice properties of Geometric Brownian Motion (few parameters, homogeneity) is a great incentive to use it in real options applications. • Pindyck (1999) wrote: “the GBM assumption is unlikely to lead to large errors in the optimal investment rule”

Mean-Reversion + Jump: the Sample Paths • 100 sample paths for mean-reversion + jumps (l = 1 jump each 5years) Jump-Reversion Sample Paths

Nominal Prices for Brent and Similar Oils (1970-2003) Jumps-up Jumps-down • By using a long-term scale, we see oil prices jumps in both directions, depending on the kind of abnormal news: jumps-up in 1973/4, 1978/9, 1990, 1999, 2002; and jumps-down in 1986, 1991, 1997, 2001

Mean-Reversion + Jumps: Dias & Rocha • We (Dias & Rocha, 1998/9) adapt the Merton´s (1976) jump-diffusion idea for the oil prices case, considering: • Normal news cause only marginal adjustment in oil prices, modeled with the continuous-time process of mean-reversion • Abnormal rare news (war, OPEC surprises, ...) cause abnormal adjustment (jumps) in petroleum prices, modeled with a Poisson process (jumps-up & jumps-down occurring in discrete-time) • A similar process of mean-reversion with jumps was used by Dias for the equity design (US$ 200 million) of the Project Finance of Marlim Field (oil prices-linked spread) • Win-win deal (higher oil prices  higher spread, and vice versa) • 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 • The Brazilian petroleum sector opening started in 1997, breaking the Petrobras’ monopoly. For E&P case: • Fiscal regime of concessions, with first-price sealed bid (like USA) • 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) • The extendible feature occurred also in USA (5 + 3 years, for some areas of GoM) and in Europe (see paper of Kemna, 1993) • Options withextendible maturitieswas studied by Longstaff (1990) for financial applications • The timing for exploratory phase (time to expiration for the development rights) was object of a public debate • The National 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) t = 0 toT1:First Period T1: First Expiration T1 to T2:Second Period T2: Second Expiration Period Available Options [Develop Now] or [Wait and See] [Develop Now] or [Extend (commit K)] or [Give-up (Return to Government)] T I M E [Develop Now] or [Wait and See] [Develop Now] or [Give-up (Return to Government)]

Extendible Option Payoff at the First Expiration • At the first expiration (T1), the firm can develop the field, or extend the option, or give-up/back to National Agency • For geometric Brownian motion, the payoff at T1 is:

Debate of Timing of Petroleum Policy • The oil companies considered very short the time of 3 + 2 years that appeared in the first draft by National Agency • It was below the international practice mainly for deepwaters areas (e.g., USA/GoM: some areas 5 + 3 years; others 10 years) • During 1998 and part of 1999, the Director of the National Petroleum Agency (ANP) insisted in this short timing policy • The numerical simulations of our paper (Dias & Rocha, 1998) concludes that the optimal timing policy should be 8 to 10 years • In January 1999 we sent our paper to the notable economist, politic and ex-Minister Delfim Netto, highlighting this conclusion • 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 • Delfim used our paper conclusions to support his view! • 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 • The table below presents the sensibility analysis for different timing policies for the petroleum sector • Option values (F) are proxy for bonus in the bid • Higher thresholds (P*) means more delay for investments • Longer timing means more bonus but more delay (tradeoff) • Table indicates a higher % gain for option value (bonus) than a % increase in thresholds (delay) • So, is reasonable to consider something between 8-10 years

PRAVAP-14: Some Real Options Projects • PRAVAP-14 is a systemic research program named Valuation of Development Projects under Uncertainties • I coordinate this systemic project by Petrobras/E&P-Corporative • I’ll present some developed real options projects: • Exploratory revelation with focus in bids (pre-PRAVAP-14) • Dynamic value of information for development projects • Selection of mutually exclusive alternatives of development investment under oil prices uncertainty (with PUC-Rio) • Analysis of alternatives of development with option to expand, considering both oil price and technical uncertainties (with PUC) • We analyze different stochastic processes and solution methods • Geometric Brownian, reversion + jumps, different mean-reversion models • Finite differences, Monte Carlo for American options, genetic algorithms • Genetic algorithms are used for optimization (thresholds curves evolution) • I call this method of evolutionary real options (I have two papers on this)

Technical Uncertainty and Value • Technical uncertainty has zero correlation with the market portfolio, then the incremental risk-premium is zero • The discount rate is the same if the project owns technical uncertainty or not, because shareholders are diversified investors • However, technical uncertainty decreases both the net present value (NPV) of the project and the real options value • Technical uncertainty almost surely will lead to exercise the wrong development project (plant capacity, safety standards, no of wells) • The sub-optimal project generates overinvestment or underinvestment when compared with the optimal investment level that maximizes NPV or ROV • Technical uncertainty can lead to exercise options when the best is not exercise the option (wait and see is better for the true value) • Technical uncertainty can lead to not exercise options when the best is to exercise the option (option deep-in-the-money for the true value) • Hence technical uncertainty decreases value due to sub-optimal decisions not because discount rate or “manager utility”

Technical Uncertainty: Threat & Opportunity Expected NPV • Technical uncertainty leads to the threat of sub-optimal development option exercise. But it is only one side of the coin. • Technical uncertainty creates also an opportunity: generates the option to invest in information before the development decision (the learning option is valuable) • This learning value will be captured by the real options model • A gamma factor willpenalize the lack of knowledge on V(B, q) • We’ll use optimal development investment equation for D(B)

Imperfect Information or Partial Revelation Case • New information reduces technical uncertainty but usually some residual uncertainty remains (partial revelation) • In this case we have three posterior distributions. For the continuous scenarios case we have infinite posterior distributions! • It is much simpler to work with the uniqueconditional expectation distribution • Soon we’ll call this conditional expectation distribution of revelation distribution

Conditional Expectation in Theory and Practice • Let us answer assignment question 1.b on the relevance of the conditional expectation concept for learning process valuation • In the last slide we saw that is much simpler to work with the unique conditional expectation distribution than many posterior distributions • Other practical advantage: expectations has a natural place in finance • Firms use current expectations to calculate the NPV or the real options exercise payoff. Ex-ante the investment in information, the new expectation is conditional. • The price of a derivative is simply an expectation of futures values (Tavella, 2002) • 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 • Even in decision analysis literature, is common to work with conditional expectation inside the maximization equation (e.g., McCardle, 1985) • But instead conditional expectation properties, the focus has been likelihood function

Information Revelation & Revelation Distribution • How model the technical uncertainty evolution after one or more (sequential) investment in information? • The process of accumulating data about a technical parameter is a learning process towards the “truth” about this parameter • It suggests the names of information revelation and revelation distribution • A similar but not equal concept is the “revelation principle” in Bayesian games that addresses the truth on a type of player. • Here the aim is revelation of the truth on a technical parameter value • When evaluating an investment in information project, the conditional expectation of the parameter X is itself a random variable E[X | I] • The distribution of conditional expectations E[X | I] is named here revelation distribution, that is, the distribution of RX = E[X | I] • The revelation distribution has nice practical properties (4 propositions) • We will use the revelation distribution in Monte Carlo simulations, in order to combine with other sources of uncertainties into a risk-neutral framework • The revelation distribution is itself a risk-neutral distribution because technical uncertainty doesn’t demand risk-premium. So, it doesn’t require risk adjustment

E&P Process and Options Oil/Gas Success Probability = p • Drill the wildcat (pioneer)? Wait? Extend? • Revelation: additional waiting incentives Expected Volume of Reserves = B Revised Volume = B’ • Appraisal phase: delineation of reserves • Technical uncertainty: sequential options • Delineated but Undeveloped Reserves. • “Wait and See”? Extend the option? Invest in information? Develop? • Developed Reserves. • Expand the production? Stop Temporally? Abandon?

Technical Uncertainty in New Basins Investment in information (wildcat drilling, etc.) RevelationDistribution Investment in information (costly and as free-rider) Possible scenarios after the information arrived during the option lease term . t = 0 t = T t = 1 Todaytechnical and economic valuation Possible scenarios after the information arrived during the first year of option term • Consider a low-explored basin, where many different oil companies will invest in wildcat drilling and seismic along the next years • Information for a specific exploration project (prospect) can be costly (our investment) or free, from the other firms investment (free-rider) • The arrival of information process leverage the option value of a tract

Valuation of Exploratory Prospect “Compact Tree” E[B] = 150 million barrels (expected reserve size) E[q] = 20% (expected quality of developed reserve) P(t = 0) = US$ 20/bbl (long-run expected price at t = 0) D(E[B]) = 200 + (2 . E[B])  D(E[B]) = 500 million $ Success 20% (CF = chance factor) Dry Hole 25 million $ (IW = wildcat investment) • Suppose that the firm has 5 years option to drill the wildcat • Other firm wants to buy the rights of the tract for $ 3 million $. • Do you sell? How valuable is the prospect?  NPV = q P B - D = (20% . 20 . 150) - 500 = + 100 MM$ However, there is only 20% chances to find petroleum EMV = Expected Monetary Value = - IW + (CF . NPV)  EMV =- 25 + (20% . 100) = - 5 million $ Do you sell the prospect rights for any offer?

Monte Carlo Combination of Uncertainties Revelation Distributions • Considering that: (a) there are a lot of uncertainties in that low known basin; and (b) many oil companies will drill wildcats in that area in the next 5 years: • The expectations in 5 years almost surely will change and so the prospect value • The revelation distributions and the risk-neutral distribution for oil prices are:

A Visual Equation for Real Options + = • 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!

E&P Process and Options Oil/Gas Success Probability = p • Drill the wildcat (pioneer)? Wait? Extend? • Revelation: additional waiting incentives Expected Volume of Reserves = B Revised Volume = B’ • Appraisal phase: delineation of reserves • Technical uncertainty: sequential options • Delineated but Undeveloped Reserves. • “Wait and See”? Invest in information? Develop? • Developed Reserves. • Expand the production? Stop Temporally? Abandon?

Investment in Information & Reduction of Uncertainty • The intuitive managerial main objective of an investment in information is to reduce the technical uncertainty (learning) • By the benefit side, the quality of a project of investment in information is related with its revelation power, that is, the capacity to reduce uncertainty with this learning project • A more expensive alternative of investment in information can be more valuable (or not) if it has a higher revelation power • We need a model that quantifies the value to reduce uncertainty in a simple way • We need to distinguish the benefits of mutually exclusivelearning projects • Can we link the expected reduction of uncertainty with the dispersion (variance) of the conditional expectation distribution? • The answer is yes and in a very simple manner! • A simple link is performed by our Proposition 3 (the paper main contribution): the expected reduction of variance is equal the variance of revelation distribution • As the volatility, the variance of revelation distribution enhances the real option value playing a key role in the dynamic value of information. We’ll see soon.

The Revelation Distribution Properties • Full revelation definition: when new information reveal all thetruth about a technical parameter X, we have full revelation • Much more common is the partial revelation case, but full revelation is important as the limit goal for any investment in information process • The revelation distributions RX (or conditional expectations distributions) have at least 4 nice properties for modeling: • Proposition 1: for the full revelation case, the distribution of revelation RX is equal to the prior (unconditional) distribution of X (RX in the limit) • Proposition 2: The expected value for the revelation distribution is equal the expected value of the original (prior) distribution • That is: E[E[X | I ]] = E[RX] = E[X] (known as law of iterated expectations) • Proposition 3: the variance of the revelation distribution is equal to the expected reduction of variance induced by the new information • Var[E[X | I ]] = Var[RX] = Var[X] - E[Var[X | I ]] = Expected Variance Reduction (this property reports the revelation power of a learning project) • 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 Area b: possible 50% chances of Bb = 100 MM bbl & 50% of nothing a b Area a: proved Ba = 100 MM bbl c d Area d: possible 50% chances of Bd = 100 MM bbl & 50% of nothing Area c: possible 50% chances of Bc = 100 MM bbl & 50% of nothing • Suppose the following stylized case of investment in information in order to get intuition on the propositions • Only one well was drilled, proving 100 MM bbl (MM = million) • Suppose there are three alternatives of investment in information (with different revelation powers): (1) drill one well (area b); (2) drill two wells (areas b + c); (3) drill three wells (b + c + d)

Visualization of Revealed Scenarios: Revelation Distribution This is exactly the prior distribution of B (Prop. 1 OK!) All the revelation distributions have the same mean (maringale): Prop. 4 OK!

Posterior Distribution x Revelation Distribution Reduction of technical uncertainty  Increase thevariance ofrevelationdistribution (and so the option value) Distributions of conditional expectations • Assignment question 1.a: Higher volatility, higher option value. Why learn?

Revelation Distribution, Experts, and NPV • This approach presents a practical way to ask the technical expert in order to evaluate an investment in information: • What is the initial uncertainty on each relevant technical parameter? That is, the mean and variance of the prior distribution • By proposition 1, the variance of the prior distribution is the variance limit for the revelation distribution generated by any learning process • By proposition 2, the revelation distribution generated by any investment in information project has the same mean of the prior distribution • For each alternative of investment in information (learning project), what is the expected reduction of variance (revelation power)? • By proposition 3, this is also the variance of the revelation distribution • Revelation power can be expressed in percentage of variance reduction • Now considere again the simple equation NPV = V - D =q B P - D • We’ll combine technical uncertainties on q and B with oil price (P) uncertainty • After an information revelation, assume that the optimal capacity choice is function only of the reserve volume:D(B) = Fixed Cost + Variable Cost x B • The capacity constrain makes E[q B] < E[q] E[B] . The factor g corrects it.

Real x Risk-Neutral Simulation • Simulation paths for the geometric Brownian motion: one using the real drift (a) and the other risk-neutral drift (r - d). • It is easy to show that the risk-neutral driftr - d = a- p where p is the risk-premium • The simulation equations are:

Oil Price Process x Revelation Process P Inv E[B] Inv • Let us answer the assignment question 1.c • Oil price (and other market uncertainties) evolves continually along the time and it is non-controllable by oil companies (non-OPEC) • Revelation distributions occur as result of events (investment in information) in discrete points along the time • For exploration of new basins sometimes the revelation of information from other firms can be relevant (free-rider), but it occurs also in discrete-time • In many cases (appraisal phase) only our investment in information is relevant and it is totally controllable by us (activate by management) • 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 this expectation can remain the same for months/years! Jump to MC slide

The Normalized Threshold and Valuation • Assignment question 1.d is about valuation under optimization • Recall that the development option is optimally exercised at the threshold V*, when V is suficiently higher than D • Exercise the option only if the project is “deep-in-the-money” • Assume the optimal investment D as a linear function of B and independent of q: D(B) = 310 + 2.1 x B (in millions $) • This means that if B varies, the exercise price D of our option also varies, and so the threshold V*. • Relevant computational time to calculate V* for different values of D • We need perform a Monte Carlo simulation to combine the uncertainties after an information revelation. • After each B sampling, it is necessary to calculate the new threshold curve V*(t) to see if the project value V = q P B is deep-in-the money • In order to reduce the computational time, we work with the normalized threshold (V/D)*. Why?

Normalized Threshold and Valuation • We will perform the valuation considering the optimal exercise at the normalized threshold level (V/D)* • After each Monte Carlo simulation combining the revelation distributions of q and B with the risk-neutral simulation of P • We calculate V = q P B and D(B), so V/D, and compare with (V/D)* • Advantage: (V/D)* is homogeneous of degree 0 in V and D. • This means that the rule (V/D)* remains valid for any V and D • So, for any revealed scenario of B, changing D, the rule (V/D)* holds • This was proved only for geometric Brownian motion • (V/D)*(t) changes only if the risk-neutral stochastic process parametersr, d, s change. But these factors don’t change at Monte Carlo simulation • The computational time of using (V/D)* is much lower than V* • The vector (V/D)*(t) is calculated only once, whereas V*(t) needs be re-calculated every iteration in the Monte Carlo simulation. • V* is a time-consuming calculus

. Real Options in Petroleum: An Overview Seminar Real Options in Real Life