Two-Factor Oil-Price Model and Real Option Valuation: An Example of Oilfield Abandonment

Transcription

1 Two-Factor Oil-Price Model and Real Option Valuation: An Example of Oilfield Abandonment B. Jafarizadeh, SPE, and R.B. Bratvold, SPE, University of Stavanger Summary We discuss the two-factor oil-price model in valuation and analysis of flexible investment decisions. In particular, we will discuss the real options formulation of a typical oilfield-abandonment problem and will apply the least-squares Monte Carlo (LSM) simulation approach for calculation of project value. In this framework, the two-factor oil-price model will go a long way in the analysis of decisions and value creation. We also propose an implied method for estimation of parameters and state variables of the two-factor price process. The method is based on implied volatility of option on futures, the shape of the forward curve, and the implicit relationship between model parameters. Introduction Over the past several decades, the oil and gas industry has adopted increasingly sophisticated methods for dealing with the uncertainties and risks embedded in the majority of their investment opportunities (Bickel and Bratvold 2008). It is recognized that a consistent probabilistic approach provides improved understanding and insights into the range of possible outcomes and their values. Yet, although most oil and gas companies appreciate the impact of commodity prices on the value of their potential investments, few are implementing price models at the level of probabilistic sophistication and realism of their, say, subsurface models. Frequently in discounted cash-flow (DCF) valuations, conservative assumptions about the price variables are used to generate information about what value could look like if things proceed poorly. The resulting corporate planning price is sometimes called the expected price, and the investment is also valued using a high and a low price. 1 This is stress testing, not valuation. Value is a price and, as such, is a number and not a distribution. Numerous oil and gas companies have made extensive use of decision-analysis methods, and some have also looked with increasing interest at recent developments in valuing the flexibility inherent in oil and gas investment opportunities. Valuing these flexibilities requires us to ask and answer some questions we usually do not address in traditional decision analysis. In decisiontree analysis, it is usually sufficient to specify a low, medium, and high scenario for the uncertain variables. Flexibility value is derived from being able to respond to uncertainties as they are being resolved and thus requires a series of conditional probability distributions. In addition to specifying a probability distribution for the price (and other uncertain variables) for the current time period, we need to specify the distribution of prices for next time period given the current prices. This allows us to determine the optimal action in any time period, given the states of the underlying uncertainties in the previous year. Most of the literature on models that try to capture the price volatility assumes that the price follows a random walk (i.e., to 1 Companies often refer to the low and high price values as the P10 and P90 value, respectively, although, clearly, they are not P10 and P90 values drawn from the underlying distribution. Copyright VC 2012 Society of Petroleum Engineers Original SPE manuscript received for review 27 October Revised manuscript received for review 17 March Paper (SPE ) peer approved 30 April consider them as stochastic processes that evolve over time) 2 (Laughton and Jacoby 1993, 1995; Cortazar and Schwartz 1994; Dixit and Pindyck 1994; Pilipovic 1998; Schwartz 1997; Schwartz and Smith 2000; Geman 2005). Clearly, a requirement for the chosen stochastic representation to be useful is that it should be consistent with the dynamics of the hydrocarbon prices over an observed time period and lead to probability distributions for the price, S t, that agrees with the observations and known characteristics of that distribution. In this paper, we illustrate the implementation and calibration of a two-factor stochastic price model developed by Schwartz and Smith (2000), hereafter referred to as the SS model. 3 This model allows mean-reversion in short-term price deviations and uncertainty in the long-term equilibrium level to which prices revert. It provides advantages over more-basic methods, but is still simple enough to be communicated to corporate decision makers who are generally not experts in financial modeling or option theory. The balance between realism and ease of communication of the model has led us to choose this model in favor of one-factor models, in which it is assumed that only one source of uncertainty contributes to the uncertainty in prices, or other multifactor models where two or more factors contribute to the uncertainty in prices (see Schwartz 1997, Geman 2000, and Cortazar and Schwartz 2003 for two- and three-factor price models). Schwartz and Smith (2000) used a Kalman filter to estimate the model parameters and state variables on the basis of historical spot and futures prices. They also mention the possibility of using implied estimates from market data about future price levels. Inspired by this, we illustrate how current market information can be used to assess the parameters and initial state variables of the SS price model. As opposed to the Kalman filter technique, the implied approach to parameter estimation is simple and intuitive, and it will generate estimates that are sufficient for most valuation assessments. We will also apply the SS price model to value the abandonment-timing option of an oil-producing field. In principle, the oil field should be abandoned as soon as the revenues from the field are less than the costs of producing its oil. However, given the uncertain nature of both the revenue and cost elements, the decision maker has to continuously evaluate the expected values of either continuing the production or abandoning the field. Modeling the abandonment-timing decisions as an American-put option and solving for the general form is complex and cumbersome (Myers and Majd 1990). Fortunately, more-recent developments in real options analysis make the value assessment easier and provide valuable decision insight. In this work, we apply the LSM approach (Longstaff and Schwartz 2001) to value the abandonment-timing option, where the oil prices follow the SS price process. 4 We 2 The fact that commodity prices are unpredictable creates a need for price modeling. In this paper we do not provide forecasting methods because it is always impossible to correctly forecast the future commodity prices. Instead, we discuss a model that is capable of appreciating the dynamics of commodity price and can create insight in the process of investment decision making. 3 Two-factor stochastic price models have also been discussed in other works (Pilipovic 1998; Baker et al. 1998). Pindyck (1999) argues that the oil prices should be modeled using a stochastic model that reverts toward a stochastically fluctuating trend line. 4 Simulation-based valuation of American options using nonparametric regression methods was initially discussed in Carriere (1996). Longstaff and Schwartz (2001) and Tsitsiklis and van Roy (1999, 2001) applied the least-squares regression for simulation-based valuation of American options. 158 July 2012 SPE Economics & Management

2 Oil Price, USD/bbl GBM - P GBM - Mean GBM - P OU - P10 OU - Mean 140 OU - P Time, years Fig. 1 Comparison of GBM and mean-reverting (OU) price processes. assume further that there is an uncertain abandonment cash flow that includes both the decommissioning costs and the potential value of selling or reusing production equipment and can thus be either negative or positive. We illustrate how the LSM method can be used to assess the value of the abandonment option and create insight into the decision. We also discuss some limitation of the LSM approach to valuation. The contributions of this paper are three-fold: (1) to familiarize the reader with the SS model and illustrate its use in valuing the abandonment option, (2) to apply the implied approach using forward curve and options on futures to estimate the parameters and state variables of the SS model, and (3) to illustrate how the LSM approach can generate decision insight for the abandonment-timing problem. The next section introduces relevant stochastic price processes and reviews some key literature. We then introduce the SS model in The Schwartz and Smith Two-Factor Price Model and Its Calibration section and illustrate the mechanics on the implied volatility parameter estimation approach. The Project Valuation section discusses project valuation, including abandonment-timing decisions, and formulates the abandonment option. In that section, we also calculate the project value using the LSM algorithm and use the SS price process for oil prices. The section concludes with analysis of the results and a discussion on potentials and weaknesses of the LSM algorithm. In the Conclusions section, we discuss some challenges. Oil-Price Modeling: An Introduction to Stochastic Price Models It is well recognized that hydrocarbon-price uncertainty is one of the main factors that drive uncertainty in economic value assessments used to make decisions in oil and gas companies. Any valuation methodology used for evaluating investment opportunities should therefore include a dynamic price model one that replicates the characteristics of real price fluctuations as a function of time, not only the mean price. There is a rich literature on oil and gas price modeling, and much of it has been motivated by the desire to improve the quality of investment valuation under price uncertainty. There have been tremendous changes in the nature of crude-oil trading over the past 30 years. Whereas major oil companies used to refine and trade the majority of their produced volumes themselves, the majority of the produced crude is now being traded in the commodity markets (Geman 2005). Oil is one of the largest commodity markets in the world, and it has evolved from trading the physical oil into a sophisticated financial market with derivative trading horizons up to 10 years or more. 5 These derivative 5 A derivative can be defined as a financial instrument whose value depends upon (or derives from) the value of other basic underlying variables. Quite often, the variables underlying derivatives are the prices of traded assets. For example, oil-price futures, forward and swaps are derivatives whose values are dependent on the traded price of oil. An option on futures contract is a derivative whose value depends on the value of a futures contract, which itself is written on oil price. contracts are now dominating the process of world-wide oil-price developments. One effect of this change is that the crude-oil markets are now liquid, global, and volatile. The early real options literature assumed that there is a single source of uncertainty related to the prices of commodities (see Brennan and Schwartz 1985 or Paddock et al for applications of single-factor price models). These studies assumed oil spot prices followed a geometric-brownian-motion (GBM) process. The GBM approach to oil-price modeling is based on an analogy with the behavior of prices of stocks in the capital markets. This price process assumes that the expected prices grow exponentially at a constant rate over time and the variance of the prices grows with proportion to time. This is the price model underlying the well-known Black-Scholes options pricing formula. The GBM price process is, however, not consistent with the behavior of commodity prices. Historically, when prices are higher than some long-run mean or equilibrium price level, more oil is supplied because the producers will have incentives to produce more and prices tend to be driven back down toward the equilibrium level. Similarly, when prices are lower than the longrun average, less oil is supplied and prices are driven back up. Therefore, although there may be short-term disequilibriums, there is a natural mean-reverting characteristic inherent to oil prices. The mean-reverting behavior of oil and gas prices has been supported in a number of studies, including the comprehensive works of Pindyck (1999, 2001). 6 The mean-reverting price process has been used for price modeling in a number of oil- and gasrelated studies (Laughton and Jacoby 1993, 1995; Cortazar and Schwartz 1994; Dixit and Pindyck 1994; Smith and McCardle 1999; Dias 2004; Begg and Smit 2007; Willigers and Bratvold 2009). 7 The effect of modeling a price process that is actually mean reverting with a GBM can be a significant overestimation of uncertainty in the resultant cash flows. This, in turn, can result in overstated option values. Fig. 1 shows a comparison of GBM and a mean-reversion process with the same volatility. As noted by several authors (Dias and Rocha 1999; Dias 2004; Geman 2005; Begg and Smit 2007), a key characteristic of oil prices is that their volatility appears to consist of normal fluctuations along with a few large jumps. These jumps are associated with the arrival of surprising or abnormal news. The most common approach to include such jumps is to combine a meanreverting process with a Poisson process, with the additional assumption that the two processes are independent. Although the one-factor model 8 can be used to capture mean reversion in the oil price, it assumes that there is no uncertainty in the long-term equilibrium price. Gibson and Schwartz (1990), Cortazar and Schwartz (1994), Schwartz (1997), Pilipovic (1998), Baker et al. (1998), Hilliard and Reis (1998), Schwartz and Smith (2000), Cortazar and Schwartz (2003), and others have introduced composite diffusions that include a second or third factor to model uncertainty explicitly in several of the price parameters. These factors include short-term deviations from the long-term equilibrium level, in the equilibrium itself, in the convenience yield, 9 or in the risk-free interest rate. Pindyck (1999) argues that the actual 6 Statistical analysis may be used to investigate whether GBM or mean-reverting processes best match the historical hydrocarbon prices. The unit root test [developed originally by Dickey and Fuller (1981)] is particularly useful for such a comparison. However, as pointed out by Dixit and Pindyck (1994), it usually requires numerous years of data to determine with any degree of confidence whether a variable (e.g., oil price) is mean reverting. For example, using approximately 30 years of oil-price/time series fails to reject the GBM hypothesis. Pindyck (1999) rejects the GBM hypothesis only after considering more than 100 years of oil-price data. 7 Uhlenbeck and Ornstein (1930) introduced the first mean-reverting model. This model has been applied in biology, physics, and other areas of study, and recently in finance, commodity derivatives pricing, and petroleum valuation, to describe the tendency of a measurement to return toward a mean level. 8 In a price model, a factor represents a market variable that exhibits some form of random behavior. The GBM and Ornstein Uhlenbeck (OU) models are one-factor models because only the price is random. In two- and three-factor models, the long-term price, convenience yield, or interest rate may be modeled as random variables in addition to the price. 9 The convenience yield represents the flow of benefits that accrue to the owner of the oil being held in storage. These benefits derive from the flexibility that is provided by having immediate access to the stored oil. July 2012 SPE Economics & Management 159

3 Oil Price, USD/bbl behavior of real prices over the past century implies that the oilprice models should incorporate mean-reversion to a stochastically fluctuating trend line. He adds that the theory of depletableresource production and pricing also confirms these findings. Schwartz (1997) compares three models of commodity prices that include mean reversion. The first of these three models was a simple one-factor model in which the logarithm of the price is assumed to follow an OU process (Uhlenbeck and Ornstein 1930). The second and third models were two-factor models. Schwartz (1997) showed that, in relative performance, the two-factor models outperformed the one-factor model for all the data sets used in the study. For an additional discussion of stochastic processes for oil prices in real options applications, see Dias (2004). The SS Two-Factor Price Model and Its Calibration In this section, we illustrate the implementation and calibration of the two-factor price process proposed by Schwartz and Smith (2000). This model allows mean reversion in short-term prices and uncertainty in the long-term equilibrium level to which prices revert. 10 The equilibrium prices are modeled as a Brownian motion, reflecting expectations of the exhaustion of existing supply, improved exploration and production technology, inflation, and political and regulatory effects. The short-term deviations from the equilibrium prices are expected to fade away in time and therefore are modeled as a mean-reverting process. These deviations reflect the short-term changes in demand, resulting from intermittent supply disruptions, and are smoothed by the ability of market participants to adjust the inventory levels in response to market conditions. The SS Model. Allow S t be the commodity price at Time t, then lnðs t Þ¼n t þ v t ;... ð1þ where n t is the long-term equilibrium price level and v t is the short-term deviation from the equilibrium prices. The long-term factor is modeled as a Brownian motion with drift rate l n and volatility r n. dn t ¼ l n dt þ r n dz n : --- Spot Prices Equilibrium Prices... ð2þ The short-term factor is modeled with a mean-reverting process with mean-reversion coefficient j 11 and volatility r v. dv t ¼ jv t dt þ r v dz v ; Expected Values P90 P Time, years Fig. 2 Confidence bands for the real stochastic process used for modeling oil prices.... ð3þ 10 The convenience yield is implicit in the SS model as opposed to the Gibson and Schwartz model, where it is modeled explicitly. 11 The mean-reversion coefficient j describes the rate at which the short-term deviations are expected to disappear. Using j, we can calculate the half-life of the deviations, [ln(2)]/ j, which is the time in which a deviation in v is expected to halve. Oil Price, USD/bbl Spot Price --- Risk-Neutral Price Time, years where dz v and dz n are correlated increments of standard Brownian-motion processes with dz n dz v ¼ q nv dt. It can be shown that the model includes the GBM and geometric OU models as special cases when there is uncertainty only in either the long-term or short-term prices, respectively. Fig. 2 shows the P10 and P90 confidence bands and the expected values for oil prices generated by the two-factor process conditional on an initial price of USD 99/bbl and market information observed on 15 May These confidence bands show that at a specific time there is a 10% and 90% chance (respectively) that the prices fall below that amount. Fig. 2 shows that the expected spot and equilibrium prices will be equal after a few years. This is caused by the fact that short-term fluctuations are expected to fade away after a few years and the only uncertainty in the spot prices would be because of the uncertainty in the equilibrium prices. This phenomenon is also consistent with the observations in the commodity markets. In these markets, the volatility of the near-maturity futures contracts is significantly higher than the volatility of far-maturity futures contracts and the trend implies that as maturity of the futures contracts increase, the volatility decreases. If we think of oil prices in terms of the twofactor price model, then we can conclude that the volatility of the near-maturity futures contracts is given by the volatility of the sum of the short-term and long-term factors. As the maturity of the futures contracts increases, the volatility approaches the volatility of the equilibrium price (Schwartz and Smith 2000). Risk-Neutral Version of the SS Process. We will calculate the project value with abandonment option using the risk-neutral valuation scheme (to be discussed in more detail in the Project Valuation section) and will thus need the risk-neutral process to describe the dynamics of the oil prices. The implied parameterestimation method is based on the use of futures contracts and options on these futures, which are valued using the risk-neutral processes. In this framework, all cash flows are calculated using the risk-neutral processes and discounted at a risk-free rate. The short-term and long-term factors in the risk-neutral version of the two-factor price process are described by the following equations: dv t ¼ð jv t k v Þdt þ r v dz v : dn t ¼ðl n k n Þdt þ r n dz n ; Expected Values P90 P10 Fig. 3 Confidence bands for the spot and risk-neutral price processes.... ð4þ... ð5þ where, as before, dz n and dz v are correlated increments of the standard Brownian motion such that dz n dz v ¼ q nvdt and where k v and k n are risk premiums that are being subtracted from the drifts of each process. 12 The risk-neutral short-term factor is now reverting to k v =j instead of zero as in the real process. The drift of the long-term factor in this model is l n ¼ l n k n. Fig The risk premium is the amount that the buyer or seller of the future contract is willing to pay in order to avoid the risk of price fluctuations. We use the standard assumption of constant market prices of risk (Schwartz 1994). 160 July 2012 SPE Economics & Management

4 compares the P10/P90 confidence bands of the spot prices and the risk-neutral prices. 13 We found it impossible to split the total estimate of l n into separate estimates of l n and k n, our calibration procedure directly estimates l n. 14 The Kalman filter (Kalman 1960) is used widely for estimating unobserved state variables and parameters (Harvey 1989; West and Harrison 1996). The Kalman filter produces estimates of a parameter on the basis of measurements that contains noise or other inaccuracies. If historical spot oil prices (values for S t ) are considered as the measurement, then because ln(s t ) ¼ n t þ v t, Kalman-filter methods can provide estimates of n t, which are normally unobservable in the market. These estimates of n t can, in turn, be used to estimate the parameters of the equation dn t ¼ l n dt þ r ndz n. 15 Note that none of the parameter-estimation approaches will produce the correct parameters, and thus there is no right method. Both the Kalman-filter approach and the implied volatility approach that we introduce in this paper provide assessments of the price model parameters. Because the methods are based on different parameters (historical futures and options vs. current futures and options), they are not directly comparable. The two approaches could conceivably be compared by looking at a specific decision situation and investigating the impact of the resulting assessments on the decision policy and its value. That is beyond the scope of this paper. Model Calibration. In the SS model, the commodity prices are mean reverting toward a stochastically fluctuating equilibrium level. This model has a total of seven parameters (j, r v, l n, r n, q v n, k v, and k n ); 13 along with two initial conditions v 0 and n 0 to be estimated. The model parameters are not observable in the commodity markets, and a standard nonlinear least-squares optimization cannot be applied directly. In the absence of observed parameters, one possible approach would be to estimate the parameters using the Kalman filter 14 (Schwartz 1997; Schwartz and Smith 2000). Another approach is to express the hidden factors in terms of the remaining model parameters and obtain an optimal fit to the observed curves (futures curve and a curve resulting from implied volatility of options on futures) at various time points. In this work, we apply the latter approach and use current spot, futures, and options on futures to calibrate the model. A key advantage to this approach is the use of the most recent market information, and the result will yield the appropriate parameters for a risk-neutral forecast of future prices. 15 In this section, we discuss and illustrate the details required to use the implied volatility approach to calibrate the SS model. In the SS price process, the short-term deviations are expected to fade away by passage of time; this means that if we could study the expected oil prices far into the future, we would observe only those price fluctuations related to the long-term factor. Intuitively, if the long maturity futures and options on futures are available for the index oil price, then the information contained in those prices can be used for estimation of parameters related to long-term factor. The volatility of long maturity futures and what is implied by the options on those futures will give information on the volatility of the Brownian-motion process that describes the dynamics of long-term factor. On the other hand, the volatility of near-term futures and what is implied by the options on those futures will give information on the mean-reverting process that describes the short-term deviations of the oil prices. The question is, How can futures and options on futures provide information from which we can assess the SS model parameters? The Black-Scholes equation used for valuation of options on stocks can be used to find the volatility of a stock given the option price by solving an inverse pricing function using, for example, Newton s method. In this work, we applied Microsoft Excel s Goal Seek function to find the value for volatility r, which makes the Black-Scholes price match the observed option price. This approach was initially applied to options on stocks, where it works because these options are freely traded in the market and where it is reasonable to use the GBM approximation to model the change over time in the underlying market value. For barrels of oil, the option is traded on the futures contracts and it means that the underlying asset for the option is a futures contract on oil price. Thus, we need to apply a different inverse model to assess the appropriate volatilities for the SS model. Schwartz and Smith (2000) argue that if we think of commodity prices in terms of their two-factor price model, the prices of the futures contracts in such a market will be log-normally distributed. The fact that market-traded futures prices are log-normally distributed allows us to write a closed-form expression for valuing European put and call options on these futures contracts. Assume F T,t is the price of a futures contract at Time t, with maturity at Time T. Following Schwartz and Smith (2000), if / ¼ ln(f T,t ) and the volatility of ln(f T,t )isr / (t,t), the value of a European call option on a futures contract maturing at Time T, with exercise Price K, and Time t until the option expires, is where c ¼ e rt ff T;0 NðdÞ KN½d r / ðt; TÞŠg;... ð6þ d ¼ lnðf=kþ r / ðt; TÞ 1=2r /ðt; TÞ:... ð7þ and N(d) indicates cumulative probabilities for the standard normal distribution [N(d) ¼ p(z < d)]. The value of a European put option with the same parameters is p ¼ e rt f F T;0 Nð dþþkn½r / ðt; TÞ dšg:... ð8þ Step 1: Estimation of r n. We recorded the prices for options on futures contracts c, prices for underlying futures contract F T,0,time to maturity T, strike prices K, and options expiry t that were reported in the New York Mercantile Exchange on 15 May The expiries of the options contracts that we used are equal to the maturity of the futures contract (i.e., T ¼ t). On the basis of these values, we solve the inverse problem to find the volatility r / (t,t) associated with each options contract. The annualized volatility (used in the simulation of prices and real options valuation) would be r / ðt:tþ p ffi t. The r / (t,t) can be written in terms of the parameters of the two-factor model: r 2 / ðt; TÞ Var½lnðF T;tÞŠ ¼ e 2jðT tþ ð1 e 2jt Þ r2 v 2j : þr 2 n þ 2e jðt tþ ð1 e jt Þ qnvr v r ð9þ n j If we again assume that the options expire at the maturity of the futures contracts, then T ¼ t, then e 2j(T t) ¼ e j(t t) ¼ 1. Furthermore, it can be shown that as the maturity of the futures contracts increases (T!1), the implied annualized volatility of the futures contracts will mostly reflect the uncertainty about the long-term factor. 16 In other words, for large T, Eq. 9 simplifies to p r / ðt; TÞ= ffiffiffi qffiffiffiffiffi T :... ð10þ r 2 n Thus, when maturities approach infinity, the implied annual volatility approaches the volatility of long-term factor r n. We can insert the implied volatility of options that expire in 6 to 8 years [i.e., r / (8, 8), r / (7, 7), or r / (6, 6) in Eq. 10] and calculate r n. Step 2: Estimation of l n and j. The diagram in Fig. 4 shows the data points obtained from the log of futures prices with various maturity dates. We have fitted a curve (the solid line) to the discrete data points observed in the market. From now on, this curve will be called the futures curve. This curve shows that the log of the futures prices is affected by short-term volatility for near-term maturity contracts, but as the time to maturity increases, the effect of the short-term fluctuations fades away. For long maturity futures, the futures curve turns into a straight line, which has the slope of l n þ 1=2r 2 n. Having estimated r n in the previous step, we can estimate l n, the risk-neutral drift rate for the longterm factor. This curve also reveals a rough estimation for the mean-reversion coefficient j. The half-life of the deviation is the length of time that the short-term deviations are expected to halve and is equal to lnð2þ=j. The curve shows that the short-term 16 ð1 e jt Þ ð1 e 2jT Þ When T ¼ t, lim ¼ 0 and lim ¼ 0. T 1 T T 1 T July 2012 SPE Economics & Management 161

5 Log of Prices Half-life = (2)/ Observed Futures Prices Slope = µ * + ½ 2 Time, years Futures Curve Fig. 4 The futures curve and its relationship with the meanreversion coefficient. Annual Volatility Volatility Curve Implied Volatility of Observed Option Prices 0 Time, years Fig. 5 The implied volatility of observed options on futures in the commodity market and the fitted volatility curve. Spot Price Log (Prices) Expected Spot Prices Slope = μ ξ + ½σ ξ 2 Deviation (χ 0 ) Short-Term Risk Premium (λχ/κ) Futures Curve Slope = μ ξ λ ξ + ½σ 2 ξ Equilibrium Price (ξ 0 ) Time Fig. 6 Relationships among parameters of the two-factor price process. deviations will decrease to half its value in approximately 10 months; this implies a mean-reversion coefficient of lnð2þ ð10=12þ ¼ 0:83. Step 3: Estimation of r v and q nv. When T ¼ t and we work with near-maturity futures contracts (T 0), it can be shown that for small T, Eq. 9 results in the annualized volatility as follows 17 : p r / ðt; TÞ= ffiffiffi qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi T ¼ e 2jT r 2 v þ r2 n T þ 2e jt q nv r n r v :... ð11þ p Eq. 11 shows that the annualized volatility [r / ðt; TÞ= ffiffiffi T ] for near-maturity futures contracts (T 0) is the sum of the volatilities of short-term and long-term factors. Because we already calculated r n in Step 1, we can now insert the implied volatility of options that expire into Eq. 10 (e.g., in 1 to 3 months, resulting in a system of two equations with two unknowns from which we can calculate r v and qn v.) Any two of r / ð1=12; 1=12Þ, r / ð1=6; 1=6Þ, or r / ð1=4; 1=4Þ, inserted into Eq. 3 will result in such a system of linear equations. Fig. 5 shows the implied volatility of observed options on futures prices taken from NYMEX on 15 May The fitted volatility curve is a helpful tool in estimating the implied volatility for far-maturity options prices and extrapolating the trend to near-maturity (T 0) options prices. 18 Note that in all our analysis, T always ¼ t because we do not have access to other combinations in the market. Step 4: Estimation of n 0, v 0, and k v. We can estimate n 0 (the long-term factor of the spot price at Time 0), v 0 (the deviation from the equilibrium at Time 0), and k v (the risk premium for the short-term factor) on the basis of the estimations performed in the previous steps and the relationships between parameters and initial state variables. In this step, the spot and futures prices provide indirect information about these unknowns. Fig. 6 depicts the relationships between the model parameters. 19 The log of the current spot oil price is the summation of n 0 and v 0 ; therefore, because S 0 is observed in the market [West Texas Intermediate (WTI) spot oil price was USD 99/bbl on 15 May 2011], we can write v 0 in terms of n 0 v 0 ¼ lnðs 0 Þ n 0 :... ð12þ Eq. 13 shows the relationship between F T,0 (price of the futures contract at Time 0 with maturity at T) and other model parameters. We have estimated l n, r n, j, r v, and q vn previously. Furthermore, on the basis of Eq. 4, we can replace v 0 with ln(s 0 ) n 0 in Eq. 5. This leaves us with two unknowns, n 0 and k v. lnðf T;0 Þ¼e jt v 0 þ n 0 þ l n T ð1 e jt Þ k v " j þ1=2 ð1 e 2jT Þ r2 v 2j þ r2 n T þ 2ð1 e jt Þ q # vnr v r n : j ð13þ 17 ð1 e 2jT Þ ð1 e jt Þ We used the fact that, when T ¼ t, lim ¼ 2j and lim ¼ j. T!0 T T!0 T 18 This is equivalent to assessing the nugget effect of a variogram, which is commonly used to calibrate geostatistical methods. As when assessing the variogram, there is no one correct value for the near-maturity options prices and its determination is subjective and based on the knowledge and experience of the assessor. 19 Note that in Fig. 6 only the data points for the lower curve (the futures curve) are observable in the market. The data points for the upper curve (the expected spot prices) are not observed in the market and therefore we cannot locate this curve in practice. As a result, we can estimate the parameters for the risk-neutral process well, but our estimates for the spot-price process will not be acceptable. 162 July 2012 SPE Economics & Management

6 TABLE 1 ANNUAL PARAMETER ESTIMATES FOR THE TWO-FACTOR OIL-PRICE MODEL WITH ASSUMED K X 50 Parameter Description Estimated Value v 0 Short-term increment of the log of the spot price 0.21 n 0 Long-term increment of the log of the spot price 4.38 r n The volatility of the long-term factor 10% r v The volatility of the short-term factor 29% l n The risk-neutral drift rate for the long-term factor 0.5% k v The risk premium for short-term factor 0 j Mean-reversion coefficient 0.83 q nv The correlation coefficient between the random increments 0.3 TABLE 2 SUMMARY OF PARAMETER-ESTIMATION PROCEDURE Step 1 - Estimation of r n Step 2 Estimation of l n and j Step 3 Estimation of r v and q nv Step 4 Estimation of n 0, v 0, and k v Calculate implied volatilities for a long-maturity futures contract using Eqs. 6 or 7; then use Eq. 9 and calculate r n. Build the log of futures curve based on the observed prices of futures contracts; calculate the slope of the log of futures curve; estimate l n by subtracting 1=2r 2 n from the slope. Estimate half-life from the futures curve; use the relationship half-life ¼ lnð2þ to estimate j. j Calculate implied volatilities for at least two near-maturity futures contracts using Eqs. 6 or 7; build a system of equations by inserting different implied volatilities in Eq. 10; insert all estimated parameters from Steps 1 and 2 into the system of equations; solve the system of equations and calculate r v and q nv. Arbitrarily set k v ¼ 0; use Eqs. 11 and 12 to build a system of equations; solve the system of equations and calculate n 0 and v 0. As discussed previously, the data points corresponding to the upper curve of Fig. 6 are not observable in the market. The risk premiums k v and k n describe the differences between the lower and upper curve, and because expected prices are not observed, these risk premiums cannot be estimated. In the risk-neutral version of the SS model, we would need only l n (which is estimated accurately using the lower curve and eliminates the need for assessing k n ). The errors in the estimate of k v shift all the estimates of n t up or down by a constant (k v =j) with v t adjusting accordingly so as to preserve the sum n t þv t corresponding to the log of expected spot price (Schwartz and Smith 2000). Assume we replace k v by k v þ D for any D, and in compensation replace v 0 by v 0 D= j and n 0 by n 0 þ D= j. These changes will not affect the risk-neutral stochastic processes. We use this property and arbitrarily set k v ¼ 0 in our estimations (which means D ¼ k v ). Then, if we use Eqs. 4 and 5 to calculate v 0 and n 0, our estimates for these initial state variables will be valid for the risk-neutral version of the SS model. We can insert the logarithm of price of a futures contract into Eq. 13, and by setting k v ¼ 0, we can estimate n 0 and v 0 using Eqs. 12 and 13. The estimated parameters used in our economic analysis are shown in Table 1. The parameter-estimation procedure, the order of the estimates, and the relationships used are summarized in Table 2. Project Valuation Introduction. In the classical DCF approach to valuation, the net present value (NPV) of a project is calculated by discounting the future expected values using a discount rate that reflects the cost of capital and desired rate of return. This discount rate is markedly higher than the prevailing risk-free interest rate and hence can be viewed as a risk-adjusted discount rate. Furthermore, most corporations use a single discount rate in the analysis of all projects or, at best, establish different discount rates for only a few large classes (e.g., political, pipeline installation vs. new field development) of investment decisions. This one-size-fits-all approach to dealing with projects mimics the risks of the overall firm, but it fails to reflect the variety of projects that feature different types and levels of uncertainty. Furthermore, using riskadjusted discount rates often leads to an undervaluation of numerous oil and gas projects with long-time horizons. Another major criticism of the classical DCF approach is that it is based on a static view, in which future decisions are assumed to depend only on information available now and not on additional information that would be available when the decision is made. This ignores management flexibility and the value that is generated by management s ability to make decisions during the execution of the project. The value of this flexibility can only be determined using a real option valuation approach. 20 Decision-tree analysis can be used to model flexibilities and options associated with oil and gas projects. Decisions to maximize the value of the project can be included as downstream decision nodes, allowing the managers to respond as uncertainties are resolved over the project s life. The resulting decision tree can then be solved using the same risk-adjusted discount rate judged to be appropriate for the original project without flexibility. Unfortunately, although this naïve approach may provide a representative model of the project and its management, it does not result in a correct valuation of the real options associated with the project. This is because the optimization that occurs at each downstream decision node changes the expected future cash flow of the project. This, in turn, changes the risk characteristics of the project because the standard deviation of the project without the flexibility is different from that of the project with the flexibility Kulatilaka (1995) argues that the real options value reflects the enhanced ability of a firm to cope with exogenous uncertainty (e.g., uncertainty in prices) and that the value of flexibility, being a more-general concept, refers to the value that is obtained by revising decisions when uncertainties resolve. In this paper, we use the terms real options and value of flexibility interchangeably when we discuss situations where value can be created by making decisions at the face of resolving uncertainties. 21 In practice, the difficulties of evaluating any investment opportunity using a single riskadjusted discount rate render the valuation method inconsistent and biased. First, the risks in a project are not distributed evenly; some variables that affect the cash flows are more risky than others and we need to have a reliable picture of the aggregate risks and uncertainties in order to apply a single risk-adjusted discount rate to all cash flows. This aggregate picture of project s risks is usually difficult to define because of the interactions between uncertainties. Second, the risks of a project are likely to change over time and applying a single risk-adjusted discount rate for valuation does not take this into account. Third, if there are options (downstream decisions) in a project, exercising these options will affect the aggregate risk profile of the project. Using a single risk-adjusted discount rate without considering this effect will result in evaluation biases. July 2012 SPE Economics & Management 163

7 To adjust the naïve approach, we can use a risk-neutral pricing approach (Smith and Nau 1995; Smith and McCardle 1999) by distinguishing between market risks, which can be hedged by trading securities, and private risks, which are project-specific risks. The risk-neutral approach provides a single, coherent risk-neutral model, which is used to estimate the value of the project both with and without options. In this approach, the probabilities or processes associated with the uncertainties or stochastic factors in the model (e.g., oil prices) are risk-adjusted. The value of the investment is then calculated by determining its expected NPV using the risk adjusted probabilities or processes for market risks and true probabilities for private risks all discounted at the risk-free rate. As will be discussed in more detail in Appendix A, risks that fall somewhere between market and private (e.g., rig-rate risks) are assessed as true probabilities conditional on the concurrent market state. Combining decision-tree analysis with the risk-neutral approach provides a consistent and correct approach to the valuation of flexible projects. Unfortunately, decision trees (or binomial lattices) are awkward to use when the investment decision is a function of multiple uncertainties and involves multiple decision points. The LSM approach is well suited for such situations because it does not suffer from the curse of dimensionality with regard to the number of uncertainties and decision points (Longstaff and Schwartz 2001; Willigers and Bratvold 2009). Using the LSM approach, we start by building a Monte Carlo simulation model that takes into account all relevant uncertainties of the problem. From this model, we generate a large number of possible outcomes for the project without options. In order to calculate an optimal exercise policy at each decision point, we need to evaluate the expected future payoff (the continuation value) for each possible choice, conditioned on the resolution of all the uncertainties up to that time. The optimal policy is then achieved by selecting the option that yields the highest continuation value given the information available. The challenge is to determine the continuation value, and this is where Longstaff and Schwartz (2001) suggest the use of linear regression. The LSM method is applied to find an approximate value function that relates the continuation value to the underlying uncertainties. Once the value function has been established, the estimated continuation value is determined by the realized values of the uncertainties in a given period. 22 Because of the long time span of numerous oil and gas projects, the abandonment decision will be made well into the future and the economic impact of this decision is not considered explicitly in the development decision because of its insignificant impact on the value of the projects at initiation. In contrast, in the final years of production, the economic impact of the abandonment decision is significant. Abandonment-Timing Option. The timing of the decision to abandon an oil or gas field can have a significant impact on asset value. As a simple rule, when the revenues from selling the extracted hydrocarbons are not sufficient to cover expenses (including tax) then the field is abandoned. In reality, the uncertain oil or gas prices, complex cash-flow structures, and interrelated decisions transform the timing of field abandonment into a complex real option. Most major oil and gas fields produce for 20 to 30 years or more before their production rates decline below an economic limit and continuing the operations would incur a net loss to the company. At that time, the decision of how and when to terminate the operations will create an important exit option. The project value is the expected sum of cash flows, including those associated with any inherent options. In some projects, the abandonment option may have a significant impact on project value. 22 After the production is closed down, the installations must be dismantled and removed. Generally, in offshore installations the topsides are taken to shore for recycling; the substructures may also be removed or left in place either temporarily or permanently (subject to regulations and government policies). Dismantling and removal of large platforms is costly and involves regulatory and environmental considerations (Osmundsen and Tveterås 2003). The removal costs are, however, highly uncertain partly because of the challenging nature of the job and partly because of the industry s lack of experience in dismantling large offshore construction (Parente et al. 2006). There are more than 6,500 offshore installations worldwide, and Osmundsen and Tveterås (2003) estimated that the cost of removal would exceed USD 20 billion. Thus, significant value loss or gain potential is associated with the timing of the abandonment. Given the uncertain future oil prices, the net cash flows from oil production will also be uncertain. In this situation, the abandonment decision must be based on the expectation of the future cash flows, given continued production. The expected future oil prices and cash flows should be conditional on the current price, and we call the conditional expectation of the future cash flows the continuation value, which includes any value gained from optimal decisions in the future. In uncertain situations, the optimal course of action would be to compare the value from immediate abandonment with the continuation value and choose the alternative with the largest value. This view of abandonment decisions as an American option is discussed in Myers and Majd (1990) and Berger et al. (1996). At the time of abandonment, the wells must be plugged and the infrastructures dismantled. In most fiscal regimes, the operating company is responsible for minimizing the risk of pollution and keeping the production site in an environmentally acceptable state. Performing this is costly, the resale value of the existing equipment is often insignificant, and thus the abandonment of a field generally requires a significant outlay for the operator. Osmundsen and Tveterås (2003) studied the decommissioning costs and policy issues on the Norwegian continental shelf. Their study reveals numerous sources of uncertainty that impact the cost of abandoning a field. In some situations, the field may be acquired by another operator 23 or there may be a positive resale value for the facilities and infrastructure. 24 In these situations, the optimal time of abandonment depends on the future oil prices, decommissioning costs, and the salvage value of the equipment, which are all uncertain. As an example, consider a field with a floating production, storage, and offloading (FPSO) vessel. At the time of the abandonment, the FPSO vessel can be refitted and reused for another field. Compared with a field that has been developed using fixed production platforms, the FPSO is a more-flexible asset and may have a positive salvage value. The salvage value may induce a different course of action and a different value for the abandonment option compared with a fixed-platform operation. Everything else being equal, flexible equipment that has a secondary market (compared with custom-built equipment that does not have a secondary market) increases the abandonment-option value (Myers and Majd 1990; Berger et al. 1996). Assume that an operator wants to assess the current value of the project including the abandonment-timing option. Such a value estimate may, for example, be relevant in negotiating a price for the transfer of the operatorship to another party. The production is currently at 1 million bbl/yr and declining. This production level is achieved by spending USD 10 million/yr in the form of fixed costs and variable operating costs that, for this project, are well estimated by average annual oil price 2=5 production rate: ð14þ 23 Although a company may abandon a field, the life of the field may not stop at that time. In some cases, smaller companies with lower overhead costs or lower profit expectations may take over the field and continue producing from it. In other cases, the national oil company of the country involved may be interested to take back the field. In these situations, the two parties can negotiate a transfer price, which is comparable to the project value with abandonment option in our analysis. 24 The emergence of new technologies or discovery of reserves in adjacent areas may affect the value of facilities because they can still be used for extraction purposes. See Osmundsen and Tveterås (2003) for a study of policy issues about decommissioning the production facilities. 25 The cost structure in a project can be far more complex than the simple cost formula used in this paper. However, we omitted lengthy discussions on the cost structure and its correlation with other value drivers in order to focus on the valuation part. In another simplifying assumption, we used the end-of-year convention for cash flows (i.e., we assume all cash flows occur at the end of each year as a lump sum). 164 July 2012 SPE Economics & Management

8 TABLE 3 YEARLY PARAMETERS FOR THE MEAN-REVERTING PROCESS DESCRIBING THE DYNAMICS OF THE ABANDONMENT CASH FLOWS Description Parameter Value Mean-Reversion Coefficient a 1 Mean Abandonment Cash Flow h USD 30 Million Volatility of Abandonment Cash Flows r h USD 9 million/yr (30% annual volatility) If we choose to continue the production in the next year, the USD 1 million will be expensed irrespective of the amount of production. However, the operating cost varies according to the level of production and includes the cost of pumping the liquids to the surface, primary treatments, separation, and transporting the oil to the supply point. The field is in late life, and the annual production decline has for several years been stable at 10% per year. Historically, a good estimate of the variable cost in each year has been given. For the purpose of this paper, we combine the decommissioning costs and salvage values into a single uncertain variable, which we call the abandonment cash flow, h t, which includes all decommissioning costs and the resale value of the FPSO and thus may take on positive or negative values. Assume the example oil field discussed earlier is producing through an FPSO that has a positive value in a secondary market. The operator expects that the abandonment cash flow, h t, of the project is time dependent and can be modeled by an OU process as follows: dh ¼ aðh hþdt þ r h dz h ;... ð15þ where a is the mean-reversion rate and r h represents the volatility of the abandonment cash flow in which the entire abandonment cash flow is assumed to occur in the year the field is abandoned. 26 The operator realizes that the salvage value of the FPSO is correlated with the oil prices, while the decommissioning costs are assumed to be uncorrelated with the prices. Therefore, h t includes risks that fall somewhere between the notions of private and market risks. Willigers (2009) looked at historical rig data and found the correlation between oil price and rig rental rates to be approximately 0.8. Because only a fraction of h t is correlated with the oil prices, 27 the operator decides to use a correlation coefficient of 0.2 for this uncertainty (i.e., q Sh ¼ 0.2). The other parameters of the OU process are shown in Table 3. The LSM Method. Although the literature recognizes the importance of uncertainty in abandonment costs and salvage values (Myers and Majd 1990), valuing flexibility in light of multiple uncertainties is difficult using traditional option valuation tools. With two uncertainties (state variables), one or two trinomial trees are required and any stochastic process beyond GBM induces timestep-dependent up and down probabilities (Hahn and Dyer 2008). Going beyond two state variables generally requires some form of simulation. In this subsection, we illustrate how the LSM method (Longstaff and Schwartz 2001) can incorporate the key uncertainties (future oil prices, which themselves consists of two underlying state variables, and abandonment cash flows) into the abandonment-timing option, where the uncertainties are treated in a more-realistic manner than by using GBM processes. The LSM approach to option valuation starts by building a Monte Carlo simulation model that takes into account all relevant uncertainties of the problem. Using this model, we generate a large number of possible outcomes for the continue to produce situation. In order to calculate the optimal abandonment policy at each decision point, we need to evaluate the expected future payoff (the continuation value) for each possible choice (abandon or continue production), conditioned on the resolution of all the uncertainties (oil price and abandonment cash flow) up to that time. The optimal policy is then achieved by selecting the option that yields the highest value given the information available. The challenge is to determine the continuation value, and this is where Longstaff and Schwartz (2001) suggested the use of linear regression. The LSM method is applied to find an approximation to the conditional value expectation given the realized, or current, values of the uncertainties in a given time period. In this work, we want to assess the value of the flexibility to abandon the field at any given point in time. The optimal abandonment time will be assessed on the basis of the information we have about the production level, decline rate, oil-price dynamics, and abandonment-cash-flow fluctuations. In our analysis, the cash flows before 0 are treated as sunk costs and will not affect any calculations. We assume the field must be abandoned within the next 15 years. 28 The cash flow at any given year depends on the (average) oil price on that year, S t ; the abandonment cash flow of the project, h t ; and the decision, d t, to abandon or to continue the project. We denote the payoff in the period (t, t þ Dt) asp(s t, h t,d t,t). We denote the present value of all future cash flows (on the basis of optimal decisions and discounted at the risk-free discount rate, r) asf(s t, h t,t). The optimal decision at t is to choose the course of action that maximizes the present value of current cash flows along with the expected cash flows from optimal decisions in the future. This is shown as the following dynamic programming equation: 29 PðS t ; h t ; d t ; tþþe rdt E FðS t ; h t ; tþ ¼max ;... ð16þ d ½FðS tþdt ; h tþdt ; t þ DtÞŠ where E() is the expectation operator. We use the integrated riskneutral valuation procedure described in Smith and Nau (1995). The oil price is treated as a market uncertainty, which can be hedged in the commodity markets using suitable market instruments. The abandonment cash flow is a technical uncertainty, but obviously has a correlation with oil prices. In our model, we determine the expected yearly cash flows by calculating trajectories for oil prices and abandonment cash flows using the risk-neutral processes. All discounting is performed at a risk-free rate of 3%. See Appendix A for further discussion on risk-neutral valuation. Because we will use Monte Carlo simulation to generate prices and cash flows, we need to discretize the SS processes and the mean-reverting process that describes the abandonment cash flow. The following time-discrete equations have been used. lnðs t Þ¼v t þ n t :... ð17þ 26 In order to keep the example simple, we assume all tax benefits or expenses are already included in ht. 27 In our model, h is categorized as a private uncertainty because it cannot be hedged in the market. However, this variable is correlated with the market uncertainty (oil prices) and it will thus be represented by a process conditional on the concurrent market. Smith (2005) discusses that calculating such a conditional probability has the same effect as risk adjusting this uncertainty. In general, if the private uncertainties are correlated with market uncertainty, we should either use risk-adjusted processes for these factors or directly correlate them with the market uncertainty. 28 For practicality, we have to assume a finite time horizon for this valuation problem. The choice of the time horizon depends on analysis needs and should be realistic for the problem. In this paper, we also assume the value-maximizing decisions are made on the basis of expected values. 29 Also known as the Bellman equation, this equation is the necessary condition for optimality associated with our dynamic programming model. See Bertsekas (2005) for an introduction to dynamic programming. July 2012 SPE Economics & Management 165

9 Frequency 2,500 2,000 1,500 1, Project value, USD Million Abandonment Cash Flow, θ 2, USD Million Abandon the Oil Field Continue Operation Second Oil Price, S 2 (USD/bbl) Fig. 7 Distribution of the project outcomes. Fig. 8 Decision map for 2. The discretization for the long-term component, Eq. 5, is n t ¼ n t 1 þ l n Dt þ r p ffiffiffiffiffi ne n Dt :... ð18þ Eq. 4 is the continuous time version of the stationary firstorder autoregressive process in discrete time (Dixit and Pindyck 1994). The exact discrete model (valid for large Dt) for the riskneutral short-term component process is v t ¼ v t 1 e jdt ð1 e jdt Þ k rffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi v j þ r 1 e ve 2 jdt v ; 2j ð19þ where e n and e v in Eqs. 18 and 19 are standard normal random variables and are correlated in each time period with the correlation coefficient q nv. It can be shown (Wiersema 2008) that if e n and e are two independent normal random variables, we can correlate the price processes in Eqs. 18 and 19 by constructing the random variable for the second process as qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi e v ¼ e n q nv þ e 1 q 2 nv:... ð20þ The risk-neutral process used for simulation of abandonment cash flows is rffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi h t ¼ h t 1 e adt þ hð1 e adt 1 e Þþr h e 2aDt h :... ð21þ 2a Again e h is the standard normal random variable. As we discussed in the preceding subsection, the uncertainty about the abandonment cash flow is technical uncertainty but has a correlation with the oil prices. We used Eq. 20 to correlate the abandonment cash flow with the long-term factor of the oil prices. We can solve this dynamic programming problem by starting at the final date T ¼ 15 and then working backward to the current date by solving the associated Bellman equation at all the decision dates t ¼ 15,, 1. Our decisions should take into account the expectation of the uncertainties about future time Probability of Abandonment periods. It is difficult to directly assess the conditional expected continuation value E[F(S tþdt, V tþdt, t þ Dt)], and, as discussed next, we approximate this by a regression function. In this paper, we numerically solve the dynamic-programming model using the LSM method. The LSM algorithm uses the regression equation as the estimator of the future values conditional on the current information (the conditional expectation value). In this regression, the current state variables (simulated oil prices and abandonment cash flows) are the independent variables. The decision outcomes (cash flows received in the future) are then used to predict the dependent variable. With this regression equation, one can simply insert current state values and obtain a prediction for the future values. We ran a simulation of 10,000 trials and determined the nearoptimal decisions in each year (the details of the LSM implementation are discussed in Appendix B). Using these near-optimal decision policies, we found an expected NPV of USD million for the project. Analysis of Results and Discussion. The option value calculated in the preceding subsection is an expected value and is based on our modeling of the underlying uncertainties. While this number gives a positive option-value estimate, it may be useful to also look at the range of possible outcomes and associated frequencies. In Fig. 7, the output of the LSM simulation has been presented in the form of a frequency diagram. This figure shows that on the basis of the results of this simulation, some scenarios result in NPV values higher or lower than USD million. It would also be beneficial to know what the optimal decision is on the basis of the value of the state variables. For example, if in 2 the oil price is S 2 and the abandonment cash flow is h 2, then what would be the optimal decision? For what range of the state variables should the decision maker abandon the project and for what range should the operations continue? Fig. 8 shows a policy map that can provide insight for these questions. Fig. 9 shows the outputs of the LSM algorithm presented in a different format. This diagram shows the probability that the project is abandoned in each year, given the information available in 0. Such presentations can provide valuable insight for the decision makers. As an example, this diagram reveals that in 5,519 iterations out of 10,000 iterations, the optimal decision was to abandon the project in the first year. 30 We can compare the value of the abandonment-timing option with the case in which the field has to be abandoned no later than a particular time (when, for example, contracts or authorities force the company effectively to abandon the field no later than a particular time). Fig. 10 shows the market value of the project for a range of different levels of operating flexibility. The horizontal axis shows the latest possible time for the project to be abandoned. For example 2 means the project can be abandoned in either 1 or 2. Fig. 9 Probability of abandoning the project in each year, given the information available at Note that Fig. 9 shows the probabilities for abandoning the field conditional on the information in 0. The decision maker may also want to know the probabilities of abandoning the field in a specific year, conditional on not having abandoned the field in previous years. This information can be also extracted from the outputs of the LSM algorithm. 166 July 2012 SPE Economics & Management

10 Project Value, USD Million Degree of Flexibility Fig. 10 Project value for different ranges of operating flexibilities. Clearly, from Fig. 10, increased flexibility with regard to the abandonment time can add significant value. For this example, the value increase is largest in the first years and then levels off at approximately 7. Conclusions In this paper, we have illustrated the details of the parameterization, using the implied volatility approach with futures and options data, and implementation of the SS two-parameter stochastic price process. We then implemented a real option valuation model for an abandonment option using the LSM formulation and the SS price process. The real options paradigm recognizes the value-creation potential resulting from decision makers active management of their investments over time. Modeling the price uncertainty using a two-factor stochastic price process provides a realistic implementation of oil-price dynamics and, thus, a realistic option value. The LSM method implemented here readily generalizes to more-complex problems. We used a second-degree polynomial equation in our regression to estimate the conditional expected value of continuation in our example. There are other problems in which using a simple regression function is unsuitable [see Stentoft (2004) for discussion on the convergence of this estimator to the true conditional expectation value]. In general, we can add any number of terms, including nonlinear terms to the regression equation as required by the problem context. The complexity of the regression equation does not impose limitations on the efficiency of the LSM algorithm, and the LSM algorithm is relatively insensitive to the number of uncertainties in the problem, but its complexity grows with the number of decision points and the number of alternatives at each decision point. We also illustrated an implied approach for calibration of the SS model to market data. This method of parameter estimation is relatively simple and practical, relies on forward market data as opposed to historical data, and should encourage analysts and decision makers to include realistic models of oil-price uncertainties in their valuation efforts. It should be noted that there is uncertainty or variability in the model parameters because the calibration is based on futures prices that change on a frequent basis. If this is a concern, it is possible to use average values or values determined by the decision maker (e.g., the chief financial officer for numerous oil companies). For a better understanding of the impact of this uncertainty, a sensitivity analysis of project values and associated decisions based on the variability of model parameters should always be conducted. In this paper, we analyzed a generic example of project abandonment without considering any tax effect. Clearly, analyzing the abandonment option under different tax regimes may have a significant impact on the optimal abandonment time. Acknowledgments We thank James E. Smith for his response to our questions. We also thank the two anonymous reviewers and Associate Editor Steve Begg, whose valuable comments, suggestions, and insights helped improve this paper. References Baker, M.P., Mayfield, E.S., and Parsons, J.E Alternative Models of Uncertainty Commodity Prices for Use with Modern Asset Pricing Methods. 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11 Harrison, J.M. and Pliska, S.R Martingales and stochastic integrals in the theory of continuous trading. Stochastic Processes and their Applications 11 (3): Harvey, A.C Forecasting, Structural Time Series Models and the Kalman Filter. Cambridge, UK: Cambridge University PressCambridge University Press. Hilliard, J.E. and Reis, J Valuation of Commodity Futures and Options Under Stochastic Convenience Yields, Interest Rates, and Jump Diffusions in the Spot. The Journal of Financial and Quantitative Analysis 33 (1): Kalman, R.E A New Approach to Linear Filtering and Prediction Problems. Journal of Basic Engineering 82 (1): dx.doi.org/ / Kulatilaka, N The Value of Flexibility: A General Model of Real Options. In Real Options in Capital Investment: Models, Strategies, and Applications, ed. L. Trigeorgis, Part II-5, Westport, Connecticut: Praeger Publishers. Laughton, D.G. and Jacoby, H.D Reversion, Timing Options, and Long-Term Decision-Making. Financial Management 22 (3): Laughton, D.G. and Jacoby, H.D The Effects of Reversion on Commodity Projects of Different Length. In Real Options in Capital Investment: Models, Strategies, and Applications, third printing edition, ed. L. Trigeorgis, Part IV, Westport, Connecticut: Praeger Publishers. Longstaff, F.A. and Schwartz, E.S Valuing American options by simulation: A simple least-squares approach. Rev. Financ. Stud. 14 (1): Luenberger, D.G Investment Science. New York: Oxford University Press. Merton, R.C An Intertemporal Capital Asset Pricing Model. Econometrica 41 (5): Myers, S.C. and Majd, S Abandonment Value and Project Life. Advances in Futures and Options Research 4: Osmundsen, P. and Tveterås, R Decommissioning of petroleum installations major policy issues. Energy Policy 31 (15): Paddock, J.L., Siegel, D.R., and Smith, J.L Option Valuation of Claims on Real Assets: The Case of Offshore Petroleum Leases. The Quarterly Journal of Economics 103 (3): / Parente, V., Ferreira, D., Moutinho dos Santos, E. et al Offshore decommissioning issues: Deductibility and transferability. Energy Policy 34 (15): Pilipovic, D Energy Risk: Valuing and Managing Energy Derivatives. New York: McGraw-Hill. Pindyck, R.S The Long-Run Evolution of Energy Prices. The Energy Journal 20 (2): Pindyck, R.S The Dynamics of Commodity Spot and Futures Markets: A Primer. The Energy Journal 22 (3). Schwartz, E.S Review of Investment Under Uncertainty, Dixit A.K. and Pindyck R.S. The Journal of Finance 49 (5): Schwartz, E.S The Stochastic Behavior of Commodity Prices: Implications for Valuation and Hedging. The Journal of Finance 52 (3): Schwartz, E. and Smith, J.E Short-Term Variations and Long-Term Dynamics in Commodity Prices. Management Science 46 (7): Smith, J.E. and McCardle, K.F Options in the Real World: Lessons Learned in Evaluating Oil and Gas Investments. Oper. Res. 47 (1): Smith, J.E. and Nau, R.F Valuing Risky Projects: Option Pricing Theory and Decision Analysis. Management Science 41 (5): Smith, J.E Alternative Approaches for Solving Real Options Problems (Comment on Brandão et al. 2005). Decision Analysis 2 (2): Stentoft, L Convergence of Least Squares Monte Carlo Approach to American Option Valuation. Management Science 50 (9). Tsitsiklis, J.N. and van Roy, B Optimal stopping of Markov processes: Hilbert space theory, approximation algorithms, and an application to pricing high-dimensional financial derivatives IEEE Transactions on Automatic Control 44 (10): dx.doi.org/ / Tsitsiklis, J.N. and van Roy, B Regression methods for pricing complex American-style options IEEE Trans. Neural Networks 12 (4): Uhlenbeck, G.E. and Ornstein, L.S On the Theory of the Brownian Motion. Physical Review 36 (5): PhysRev West, M. and Harrison, J Bayesian Forecasting and Dynamic Models, second edition. New York: Springer-Verlag. Wiersema, U.F Brownian Motion Calculus. West Sussex, UK: John Wiley & Sons. Willigers, B.J.A Practical Portfolio Simulation: Determining the Precision of a Probability Distribution of a Large Asset Portfolio when the Underlying Project Economics are Captured by a Small Number of Discrete Scenarios. Paper SPE presented at the SPE Annual Technical Conference and Exhibition, New Orleans, 4 7 October. Willigers, B.J.A. and Bratvold, R.B Valuing Oil and Gas Options by Least-Squares Monte Carlo Simulation. SPE Proj Fac & Const 4 (4): SPE PA. Appendix A: Risk-Neutral Valuation of Uncertainties That Fall Between Market and Private Uncertainties Traditional DCF analysis considers uncertainty in cash flows as a source of risk. In order to reflect the effects of risk on the project value, the common approach has been to risk adjust the discount rate. In performing this, we make a number of assumptions. First, when we risk adjust the discount rate, we treat all cash flows (and elements of cash flows subject to various sources of risk) equally, as if the risk is distributed uniformly across all elements of the cash flows. 31 Second, using a single risk-adjusted discount rate for valuations also implies a constant risk level over time, whereas the risks of a project usually vary with time. 32 Third, when the risk-adjusted discount rates are used in decision trees or dynamic programs, at each decision point the risk characteristics of a project will change as a result of the optimization. This will result in an underestimation of project value. In the risk-neutral valuation technique (Cox et al. 1985), 33 the value of an investment in a complete market is the discounted expected value of its future payoffs under the risk-neutral probabilities. In other words, we risk adjust probabilities and stochastic processes associated with uncertainties in the model. Then, we calculate the expected NPV of the investment using these riskneutral probabilities or processes and discount them at the riskfree rate. On the basis of the equilibrium risk-neutral scheme described in Cox et al. (1985), the risk premium k i for uncertain Variable i on the basis of Merton s (1973) intertemporal capital asset pricing model is a function of r im (the covariance between the uncertain factor and the market), r 2 m (the variance of the market), r (the risk-free rate), and r m (the expected return on the market portfolio), following the equation r 2 m k i ¼ r im r 2 ðr m rþ:... ða-1þ m In Eq. A-1, r im can be regarded as beta of the uncertain Variable i. Such a risk premium can be used, for example, to risk adjust the oil-price process by deducting it from the drift rate of 31 In most projects, the elements of cash flows have different degrees of uncertainty. For example, royalties, tariffs, or booking costs are usually significantly less uncertain than the revenues. 32 For example, a major oil-producing field is considered significantly less risky when it is matured compared with the early development phases. In general, the uncertainty about recoverable oil volume tends to decrease with a project s maturity. 33 The risk-neutral valuation scheme for option valuation was developed in Cox and Ross (1976). Other applications of risk-neutral valuation include Cox et al. (1979) for discretetime models and Harrison and Pliska (1981) for continuous models. 168 July 2012 SPE Economics & Management

12 TABLE B-1 ANNUAL CASH FLOWS AND ABANDONMENT CASH FLOWS FOR SEVEN SAMPLE PATHS Path 1 Oil Price (USD/bbl) h (USD Million) Path 2 Oil Price (USD/bbl) h (USD Million) Path 3 Oil Price (USD/bbl) h (USD Million) Path 4 Oil Price (USD/bbl) h (USD Million) Path 5 Oil Price (USD/bbl) h (USD Million) Path 6 Oil Price (USD/bbl) h (USD Million) Path 7 Oil Price (USD/bbl) h (USD Million) the process. If an uncertain factor is uncorrelated to the market movements, then its risk premium is zero and does not need risk adjustment. All the cash flows generated using these probabilities and stochastic processes are discounted using the risk-free discount rate [see Luenberger (1998) for a discussion on risk-neutral valuation]. Smith and Nau (1995) proposed a modification to this approach in which uncertainties are divided into market uncertainties (those that can be hedged in the market by trading securities) and private uncertainties (those that cannot be hedged in the market). The expected NPV of the investment is calculated using the risk-adjusted probabilities or processes for market uncertainties and assessed probabilities for private uncertainties. The dependence between market and private risks is captured by assessing true probabilities for the private risks conditional on the contemporaneous market state (Smith 2005). We may not always be able to categorize all uncertainties into either market or private uncertainties. Some uncertainties (e.g., the uncertainty about abandonment cash flow in our example) may fall somewhere between these two categories. However, we should note that only those uncertainties that can be hedged with market instruments are considered market uncertainties. Oil-price uncertainty is an example of a market uncertainty because in major commodity markets there are futures and options on futures contracts available to hedge the oil-price uncertainty. 34 The price of other crudes produced from specific fields should be judged on the basis of their correlation with traded crude types and may not be regarded easily as market uncertainty. Those uncertainties that cannot be hedged using market instruments should be regarded as private uncertainties, even though they may be related to market uncertainties. As discussed by Smith (2005), if private uncertainty p has a correlation with market uncertainty m, then in the decision-analytic risk-neutral approach, we should use the true probabilities for p conditional on the contemporaneous state of m. If we assume p and m are correlated with correlation coefficient q pm, then the conditional stochastic process for p will have a reduction equal to b p (l m r), where r is the risk-free rate, l m is the r p return on market uncertainty m, and b p ¼ q pm ¼ r pm r m r 2 can be m interpreted as beta of uncertainty p in a fashion similar to Eq. A-1. We expect that the risk-neutral valuation scheme is a moreconsistent valuation approach in that each individual uncertainty is 34 In terms of the decision-analytic risk-neutral valuation, only a few crude-oil types (WTI crude oil traded in NYMEX, Brent crude traded in IntercontinentalExchange, and a few other crudes) are actively traded in the futures markets. The price uncertainty in these crudes can be hedged using the market instruments, and we can categorize them as market uncertainties. risk adjusted separately, whereas risk adjusting the discount rate for the aggregate project level, although possible, is prone to errors and requires harder judgments. Once the uncertainties are risk adjusted, they also can be used across different projects. Furthermore, it is more plausible to assume that risk adjustments for uncertainties stay constant over time. Finally, projects with options can be modeled more consistently using this method because changing the course of a project can alter exposure to some uncertainties but will not affect the risk characteristic of the individual factors. Appendix B: Numerical Valuation A simulation with 10,000 trials generates 10,000 paths for oil prices and abandonment cash flows from the equations offered in The LSM Method section. These values are then used in a spreadsheet to create 10,000 paths for the 0 to 15 cash flows. 35 Out of all simulated paths, seven paths are presented in Tables B-1 and B-2. These paths represent seven scenarios that will result in different abandonment decisions. For example, in Path 1, the cash-flow becomes negative in 7 because of a decrease in the oil prices and decline in the production rate. The yearly cash flow will change sign two more times before the project is abandoned in 14. The owner may abandon the project any time between s 1 and 15. If the owner decides to abandon, he or she will receive that year s cash flow and the prevailing abandonment cash flow. In some other cases, it may be better to continue the production and take advantage of higher oil prices and abandonment cash flows in the future. In our LSM implementation, we should consider these decision policies in each year and find the optimum abandonment time for each path of the simulation. The LSM algorithm cannot provide a course of action better than the decision alternatives we improvised in the first place; it is thus important to emphasize the creative thinking and come up with the value-creating decision alternatives. The LSM algorithm starts at 15 and determines the optimal decision conditional on not abandoning the project in previous years. If the oil field is still generating positive cash flows at 15, we would have no choice other than to abandon. The optimal decision at 15 would then be to abandon in all 10,000 paths. The next step would be to determine the optimal decision in 14. The cash flow in 14 is observable directly by the decision maker and can be used in the decisions. At this stage, the 35 We used a 15-year time horizon in our example because the production beyond that time was negligible. In the Conclusions section, we discussed the effect of time horizon (and number of subperiods) in the efficiency of LSM simulation. 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13 TABLE B-2 CASH-FLOW MATRIX FOR THE SEVEN SAMPLE PATHS NPV (USD Million) Path Path Path Path Path Path Path owner has to decide between abandoning at 14 and continuing production until 15. If the owner decides to abandon the project in 14, the immediate payoff would be the 14 cash flow along with the 14 abandonment cash flow. If the decision maker decides to continue production, he or she will receive the cash flows from s 14 and 15. Because information about 15 cash flows is not available at 14, the decision maker should be able to estimate the 15 cash flow on the basis of the information available at 14. Such estimation is made with the help of a regression function. The general form for such a regression function is as follows: E½FðS 15 ; V 15 ; tþš a 1 S 14 þ a 2 h 14 þ a 3 S 2 14 þ a 4h 2 14 þa 5 S 14 h 14 þ b: ðb-1þ We use a two-factor stochastic model to describe the variability of the oil prices, meaning that the oil price S t is a function of two stochastic factors n t and v t. To include all this information in the regression function, Eq. B-1 can be expanded as follows: E½FðS 15 ; V 15 ; tþš a 0 1 n 14 þ a 00 1 v 14 þ a 2 h 14 þ a 0 3 n2 14 þ a00 3 v2 14 þa 4 h 2 14 þ a 5n 14 h 14 þ a 6 v 14 h 14 þ a 7 n 14 v 14 þ b: ðb-2þ The coefficients of the regression function can be estimated by the LSM using the simulation data. The 10,000 data points from 14 oil prices and abandonment cash flows serve as the independent variables, and 10,000 data points from 15 decision outcomes serve as the dependent variables. The LSM algorithm compares immediate payoffs with the expected future payoffs in each path. If the current cash flow is positive, and the regression function estimates an expected positive payoff in the future, the LSM algorithm chooses to continue to the following year. On the other hand, if the current cash flow is negative, or the regression function estimates an expected negative payoff in the future, the LSM algorithm chooses to abandon the project. After the optimal abandonment decisions are determined for a year, the LSM algorithm moves one step back to the previous year and repeats this procedure until the optimal decisions in all years are determined. 36 The LSM algorithm is a backward recursion approach applied to multistage decision problems. At each decision point, the optimal decision alternative is determined with an eye to the evolution of the uncertainty in the future. The algorithm should also consider that the abandonment can occur only once during the 15 years. After the optimal decisions are determined for each path, the algorithm looks for the earliest abandonment date and assigns the value of zero to cash flows that occur after this date. For example, if in a path the earliest abandonment date determined by the LSM is 9, the algorithm assigns the value of zero to cash flows in s 10 through 15. Applying this process to all paths will generate the cash-flow matrix. This matrix is shown in Table B-2 for the sample paths presented in Table B-1. In the cash-flow matrix, all cash flows from 0 to the year of project abandonment are reported. To calculate the value of the project with the abandonment option, the NPV of the cash flows for each path is calculated and then averaged over all paths. Babak Jafarizadeh is currently a senior analyst in Statoil, Norway, and his research interests include real option valuation, decision analysis, and portfolio management in the oil and gas industry. He holds a PhD degree in petroleum investment and decision analysis from the University of Stavanger and an MSc degree in financial engineering from Amirkabir University of Technology (Tehran Polytechnic) in Iran. Reidar B. Bratvold is a professor of petroleum investment and decision analysis at the University of Stavanger and at the Norwegian University of Science and Technology in Trondheim, Norway. His research interests include decision analysis, representing and solving decision problems in the upstream oil and gas industry, valuation of risky projects, portfolio analysis, and behavioral challenges in decision making. Before entering academia, Bratvold spent 15 years in the industry in various technical and management roles. He is a coauthor of SPE primer Making Good Decisions. Bratvold is an associate editor for SPE Economics & Management and has twice served as an SPE Distinguished Lecturer. He was made a member of the Norwegian Academy of Technological Sciences for his work in petroleum investment and decision analysis and is also an elected Fellow with the Society of Decision Professionals. Bratvold holds a PhD degree in petroleum engineering and an MSc degree in mathematics, both from Stanford University, and has business and management-science education from INSEAD and Stanford University. 36 Clearly, the LSM algorithm is an approximate algorithm and the resulting decisions will be near-optimal, which will generate a near-optimal value for the abandonment flexibility [see Glasserman (2004) for a review of the approximation biases]. 170 July 2012 SPE Economics & Management