Two Factor Oil Price Model

Two-Factor Oil-Price Model and RealOption Valuation: An Example of Oilfield

AbandonmentB. Jafarizadeh, SPE, and R.B. Bratvold, SPE, University of Stavanger

Summary

We discuss the two-factor oil-price model in valuation and analy-sis of flexible investment decisions. In particular, we will discussthe real options formulation of a typical oilfield-abandonmentproblem and will apply the least-squares Monte Carlo (LSM) sim-ulation approach for calculation of project value. In this frame-work, the two-factor oil-price model will go a long way in theanalysis of decisions and value creation. We also propose animplied method for estimation of parameters and state variablesof the two-factor price process. The method is based on impliedvolatility of option on futures, the shape of the forward curve, andthe implicit relationship between model parameters.

Introduction

Over the past several decades, the oil and gas industry has adoptedincreasingly sophisticated methods for dealing with the uncertain-ties and risks embedded in the majority of their investment op-portunities (Bickel and Bratvold 2008). It is recognized that aconsistent probabilistic approach provides improved understand-ing and insights into the range of possible outcomes and their val-ues. Yet, although most oil and gas companies appreciate theimpact of commodity prices on the value of their potential invest-ments, few are implementing price models at the level of probabil-istic sophistication and realism of their, say, subsurface models.

Frequently in discounted cash-flow (DCF) valuations, con-servative assumptions about the price variables are used to gen-erate information about what value could look like if thingsproceed poorly. The resulting corporate planning price is some-times called the expected price, and the investment is alsovalued using a high and a low price.1 This is stress testing, notvaluation. Value is a price and, as such, is a number and not adistribution.

Numerous oil and gas companies have made extensive use ofdecision-analysis methods, and some have also looked withincreasing interest at recent developments in valuing the flexibil-ity inherent in oil and gas investment opportunities. Valuing theseflexibilities requires us to ask and answer some questions we usu-ally do not address in traditional decision analysis. In decision-tree analysis, it is usually sufficient to specify a low, medium, andhigh scenario for the uncertain variables. Flexibility value isderived from being able to respond to uncertainties as they arebeing resolved and thus requires a series of conditional probabilitydistributions. In addition to specifying a probability distributionfor the price (and other uncertain variables) for the current timeperiod, we need to specify the distribution of prices for next timeperiod given the current prices. This allows us to determine theoptimal action in any time period, given the states of the underly-ing uncertainties in the previous year.

Most of the literature on models that try to capture the pricevolatility assumes that the price follows a random walk (i.e., to

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 requirementfor the chosen stochastic representation to be useful is that itshould be consistent with the dynamics of the hydrocarbon pricesover an observed time period and lead to probability distributionsfor the price, St, that agrees with the observations and knowncharacteristics of that distribution.

In this paper, we illustrate the implementation and calibrationof a two-factor stochastic price model developed by Schwartz andSmith (2000), hereafter referred to as the SS model.3 This modelallows mean-reversion in short-term price deviations and uncer-tainty in the long-term equilibrium level to which prices revert. Itprovides advantages over more-basic methods, but is still simpleenough to be communicated to corporate decision makers who aregenerally not experts in financial modeling or option theory. Thebalance between realism and ease of communication of the modelhas led us to choose this model in favor of one-factor models, inwhich it is assumed that only one source of uncertainty contrib-utes to the uncertainty in prices, or other multifactor modelswhere two or more factors contribute to the uncertainty in prices(see Schwartz 1997, Geman 2000, and Cortazar and Schwartz2003 for two- and three-factor price models).

Schwartz and Smith (2000) used a Kalman filter to estimate themodel parameters and state variables on the basis of historical spotand futures prices. They alsomention the possibility of using impliedestimates from market data about future price levels. Inspired bythis, we illustrate how current market information can be used toassess the parameters and initial state variables of the SS pricemodel. As opposed to the Kalman filter technique, the impliedapproach to parameter estimation is simple and intuitive, and it willgenerate estimates that are sufficient for most valuation assessments.

We will also apply the SS price model to value the abandon-ment-timing option of an oil-producing field. In principle, the oilfield should be abandoned as soon as the revenues from the fieldare less than the costs of producing its oil. However, given theuncertain nature of both the revenue and cost elements, the deci-sion maker has to continuously evaluate the expected values of ei-ther continuing the production or abandoning the field. Modelingthe abandonment-timing decisions as an American-put option andsolving for the general form is complex and cumbersome (Myersand Majd 1990). Fortunately, more-recent developments in realoptions analysis make the value assessment easier and provide val-uable decision insight. In this work, we apply the LSM approach(Longstaff and Schwartz 2001) to value the abandonment-timingoption, where the oil prices follow the SS price process.4 We

1 Companies often refer to the low and high price values as the P10 and P90 value, respec-

tively, although, clearly, they are not P10 and P90 values drawn from the underlying

distribution.

CopyrightVC 2012 Society of Petroleum Engineers

Original SPE manuscript received for review 27 October 2010. Revised manuscript receivedfor review 17 March 2012. Paper (SPE 162862) peer approved 30 April 2012.

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 cor-

rectly 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.

EM162862 DOI: 10.2118/162862-PA Date: 19-July-12 Stage: Page: 158 Total Pages: 13

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158 July 2012 SPE Economics & Management

assume further that there is an uncertain abandonment cash flowthat includes both the decommissioning costs and the potentialvalue of selling or reusing production equipment and can thus beeither negative or positive. We illustrate how the LSM method canbe used to assess the value of the abandonment option and createinsight into the decision. We also discuss some limitation of theLSM approach to valuation.

The contributions of this paper are three-fold: (1) to familiar-ize the reader with the SS model and illustrate its use in valuingthe abandonment option, (2) to apply the implied approach usingforward curve and options on futures to estimate the parametersand state variables of the SS model, and (3) to illustrate how theLSM approach can generate decision insight for the abandon-ment-timing problem. The next section introduces relevant sto-chastic price processes and reviews some key literature. We thenintroduce the SS model in The Schwartz and Smith Two-FactorPrice Model and Its Calibration section and illustrate the mechan-ics on the implied volatility parameter estimation approach. TheProject Valuation section discusses project valuation, includingabandonment-timing decisions, and formulates the abandonmentoption. In that section, we also calculate the project value usingthe LSM algorithm and use the SS price process for oil prices.The section concludes with analysis of the results and a discussionon potentials and weaknesses of the LSM algorithm. In the Con-clusions section, we discuss some challenges.

Oil-Price Modeling: An Introduction to StochasticPrice Models

It is well recognized that hydrocarbon-price uncertainty is one ofthe main factors that drive uncertainty in economic value assess-ments used to make decisions in oil and gas companies. Any valu-ation methodology used for evaluating investment opportunitiesshould therefore include a dynamic price modelone that repli-cates the characteristics of real price fluctuations as a function oftime, not only the mean price. There is a rich literature on oil andgas price modeling, and much of it has been motivated by thedesire to improve the quality of investment valuation under priceuncertainty.

There have been tremendous changes in the nature of crude-oiltrading over the past 30 years. Whereas major oil companies usedto refine and trade the majority of their produced volumes them-selves, the majority of the produced crude is now being traded inthe commodity markets (Geman 2005). Oil is one of the largestcommodity markets in the world, and it has evolved from tradingthe physical oil into a sophisticated financial market with deriva-tive trading horizons up to 10 years or more.5 These derivative

contracts are now dominating the process of world-wide oil-pricedevelopments. One effect of this change is that the crude-oil mar-kets are now liquid, global, and volatile.

The early real options literature assumed that there is a singlesource of uncertainty related to the prices of commodities (seeBrennan and Schwartz 1985 or Paddock et al. 1988 for applica-tions of single-factor price models). These studies assumed oilspot prices followed a geometric-Brownian-motion (GBM) pro-cess. The GBM approach to oil-price modeling is based on an anal-ogy with the behavior of prices of stocks in the capital markets.This price process assumes that the expected prices grow exponen-tially at a constant rate over time and the variance of the pricesgrows with proportion to time. This is the price model underlyingthe well-known Black-Scholes options pricing formula.

The GBM price process is, however, not consistent with thebehavior of commodity prices. Historically, when prices arehigher than some long-run mean or equilibrium price level, moreoil is supplied because the producers will have incentives to pro-duce more and prices tend to be driven back down toward theequilibrium level. Similarly, when prices are lower than the long-run 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 pri-ces. The mean-reverting behavior of oil and gas prices has beensupported in a number of studies, including the comprehensiveworks of Pindyck (1999, 2001).6 The mean-reverting price pro-cess has been used for price modeling in a number of oil- and gas-related studies (Laughton and Jacoby 1993, 1995; Cortazar andSchwartz 1994; Dixit and Pindyck 1994; Smith and McCardle1999; Dias 2004; Begg and Smit 2007; Willigers and Bratvold2009).7 The effect of modeling a price process that is actuallymean reverting with a GBM can be a significant overestimation ofuncertainty in the resultant cash flows. This, in turn, can result inoverstated option values. Fig. 1 shows a comparison of GBM anda 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 pri-ces is that their volatility appears to consist of normal fluctua-tions along with a few large jumps. These jumps are associatedwith the arrival of surprising or abnormal news. The mostcommon approach to include such jumps is to combine a mean-reverting process with a Poisson process, with the additionalassumption that the two processes are independent.

Although the one-factor model8 can be used to capture mean re-version in the oil price, it assumes that there is no uncertainty in thelong-term equilibrium price. Gibson and Schwartz (1990), Corta-zar and Schwartz (1994), Schwartz (1997), Pilipovic (1998), Bakeret al. (1998), Hilliard and Reis (1998), Schwartz and Smith (2000),Cortazar and Schwartz (2003), and others have introduced compos-ite diffusions that include a second or third factor to model uncer-tainty explicitly in several of the price parameters. These factorsinclude short-term deviations from the long-term equilibrium level,in the equilibrium itself, in the convenience yield,9 or in therisk-free interest rate. Pindyck (1999) argues that the actual

8090

100110120130140150160

0 1 2 3 4 5 6 7 8 9 10

Oil P

rice,

USD

/bbl

Time, years

GBM - P10GBM - MeanGBM - P90OU - P10OU - MeanOU - P90

Fig. 1Comparison of GBM and mean-reverting (OU) priceprocesses.

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 under-

lying 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.

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 hy-

pothesis. 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 mea-

surement 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 OrnsteinUhlenbeck (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

behavior of real prices over the past century implies that the oil-price models should incorporate mean-reversion to a stochasticallyfluctuating trend line. He adds that the theory of depletable-resource production and pricing also confirms these findings.Schwartz (1997) compares three models of commodity prices thatinclude mean reversion. The first of these three models was a sim-ple one-factor model in which the logarithm of the price is assumedto follow an OU process (Uhlenbeck and Ornstein 1930). The sec-ond and third models were two-factor models. Schwartz (1997)showed that, in relative performance, the two-factor models out-performed the one-factor model for all the data sets used in thestudy. For an additional discussion of stochastic processes for oilprices in real options applications, see Dias (2004).

The SS Two-Factor Price Model andIts Calibration

In this section, we illustrate the implementation and calibration ofthe two-factor price process proposed by Schwartz and Smith(2000). This model allows mean reversion in short-term pricesand uncertainty in the long-term equilibrium level to which pricesrevert.10 The equilibrium prices are modeled as a Brownianmotion, reflecting expectations of the exhaustion of existing sup-ply, improved exploration and production technology, inflation,and political and regulatory effects. The short-term deviationsfrom the equilibrium prices are expected to fade away in time andtherefore are modeled as a mean-reverting process. These devia-tions reflect the short-term changes in demand, resulting fromintermittent supply disruptions, and are smoothed by the ability ofmarket participants to adjust the inventory levels in response tomarket conditions.

The SS Model. Allow St be the commodity price at Time t, then

lnSt nt vt; 1

where nt is the long-term equilibrium price level and vt is theshort-term deviation from the equilibrium prices. The long-termfactor is modeled as a Brownian motion with drift rate ln and vol-atility rn.

dnt lndt rndzn: 2

The short-term factor is modeled with a mean-reverting pro-cess with mean-reversion coefficient j11 and volatility rv.

dvt jvtdt rvdzv; 3

where dzv and dzn are correlated increments of standard Brown-ian-motion processes with dzndzv qnvdt. It can be shown that themodel includes the GBM and geometric OU models as specialcases when there is uncertainty only in either the long-term orshort-term prices, respectively.

Fig. 2 shows the P10 and P90 confidence bands and theexpected values for oil prices generated by the two-factor processconditional on an initial price of USD 99/bbl and market informa-tion observed on 15 May 2011. These confidence bands show thatat 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 willbe equal after a few years. This is caused by the fact that short-termfluctuations are expected to fade away after a few years and theonly uncertainty in the spot prices would be because of the uncer-tainty in the equilibrium prices. This phenomenon is also consistentwith the observations in the commodity markets. In these markets,the volatility of the near-maturity futures contracts is significantlyhigher than the volatility of far-maturity futures contracts and thetrend implies that as maturity of the futures contracts increase, thevolatility decreases. If we think of oil prices in terms of the two-factor price model, then we can conclude that the volatility of thenear-maturity futures contracts is given by the volatility of the sumof the short-term and long-term factors. As the maturity of thefutures contracts increases, the volatility approaches the volatilityof the equilibrium price (Schwartz and Smith 2000).

Risk-Neutral Version of the SS Process. We will calculate theproject value with abandonment option using the risk-neutral val-uation scheme (to be discussed in more detail in the Project Valu-ation section) and will thus need the risk-neutral process todescribe the dynamics of the oil prices. The implied parameter-estimation method is based on the use of futures contracts andoptions on these futures, which are valued using the risk-neutralprocesses. In this framework, all cash flows are calculated usingthe risk-neutral processes and discounted at a risk-free rate. Theshort-term and long-term factors in the risk-neutral version of thetwo-factor price process are described by the following equations:

dvt jvt kvdt rvdzv: 4

dnt ln kndt rndzn; 5

where, as before, dzn and dzv are correlated increments of the

standard Brownian motion such that dzndzv qnvdt and where

kv and kn are risk premiums that are being subtracted from thedrifts of each process.12 The risk-neutral short-term factor is nowreverting to kv=j instead of zero as in the real process. The driftof the long-term factor in this model is ln ln kn. Fig. 3

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5060708090

100110120130140150

0 1 2 3 4 5 6 7 8 9 10

Oil P

rice,

USD

/bbl

Time, years

ExpectedValues

P90

P10

- - - Spot PricesEquilibrium Prices

Fig. 2Confidence bands for the real stochastic process usedfor modeling oil prices.

5060708090

100110120130140150

0 1 2 3 4 5 6 7 8 9 10

Oil P

rice,

US

D/bb

l

Time, years

- - - Spot Price- - - Risk-Neutral Price

ExpectedValues

P90

P10

Fig. 3Confidence bands for the spot and risk-neutral priceprocesses.

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 deviationsare 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.

12 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 con-

stant market prices of risk (Schwartz 1994).




compares the P10/P90 confidence bands of the spot prices and therisk-neutral prices.

Model Calibration. In the SS model, the commodity prices aremean reverting toward a stochastically fluctuating equilibriumlevel. This model has a total of seven parameters (j, rv, ln, rn,qvn, kv, and kn);

13 along with two initial conditions v0 and n0 tobe estimated. The model parameters are not observable in thecommodity markets, and a standard nonlinear least-squares opti-mization cannot be applied directly. In the absence of observedparameters, one possible approach would be to estimate the pa-rameters using the Kalman filter14 (Schwartz 1997; Schwartz andSmith 2000). Another approach is to express the hidden factors interms of the remaining model parameters and obtain an optimal fitto the observed curves (futures curve and a curve resulting fromimplied volatility of options on futures) at various time points. Inthis work, we apply the latter approach and use current spot,futures, and options on futures to calibrate the model. A keyadvantage to this approach is the use of the most recent marketinformation, and the result will yield the appropriate parametersfor a risk-neutral forecast of future prices.15 In this section, we dis-cuss and illustrate the details required to use the implied volatilityapproach to calibrate the SSmodel.

In the SS price process, the short-term deviations are expectedto fade away by passage of time; this means that if we could studythe expected oil prices far into the future, we would observe onlythose price fluctuations related to the long-term factor. Intuitively,if the long maturity futures and options on futures are availablefor the index oil price, then the information contained in those pri-ces can be used for estimation of parameters related to long-termfactor. The volatility of long maturity futures and what is impliedby the options on those futures will give information on the vola-tility of the Brownian-motion process that describes the dynamicsof long-term factor. On the other hand, the volatility of near-termfutures and what is implied by the options on those futures willgive information on the mean-reverting process that describes theshort-term deviations of the oil prices. The question is, How canfutures and options on futures provide information from which wecan assess the SS model parameters?

The Black-Scholes equation used for valuation of options onstocks can be used to find the volatility of a stock given the optionprice by solving an inverse pricing function using, for example,Newtons method. In this work, we applied Microsoft ExcelsGoal Seek function to find the value for volatility r, which makesthe Black-Scholes price match the observed option price. Thisapproach was initially applied to options on stocks, where itworks because these options are freely traded in the market andwhere it is reasonable to use the GBM approximation to modelthe change over time in the underlying market value. For barrelsof oil, the option is traded on the futures contracts and it meansthat the underlying asset for the option is a futures contract on oilprice. Thus, we need to apply a different inverse model to assessthe appropriate volatilities for the SS model.

Schwartz and Smith (2000) argue that if we think of commod-ity prices in terms of their two-factor price model, the prices ofthe futures contracts in such a market will be log-normally distrib-

uted. The fact that market-traded futures prices are log-normallydistributed allows us to write a closed-form expression for valuingEuropean put and call options on these futures contracts. AssumeFT,t is the price of a futures contract at Time t, with maturity atTime T. Following Schwartz and Smith (2000), if / ln(FT,t) andthe volatility of ln(FT,t) is r/(t,T), the value of a European calloption on a futures contract maturing at Time T, with exercisePrice K, and Time t until the option expires, is

c ertfFT;0Nd KNd r/t;Tg; 6

where

d lnF=Kr/t;T 1=2r/t;T: 7

and N(d) indicates cumulative probabilities for the standard nor-mal distribution [N(d) p(Z< d)]. The value of a European putoption with the same parameters is

p ertfFT;0Nd KNr/t; T dg: 8

Step 1: Estimation of rn. We recorded the prices for options onfutures contracts c, prices for underlying futures contract FT,0, timeto maturity T, strike prices K, and options expiry t that were reportedin the New York Mercantile Exchange on 15 May 2011. The expi-ries of the options contracts that we used are equal to the maturity ofthe futures contract (i.e., T t). On the basis of these values, wesolve the inverse problem to find the volatility r/(t,T) associatedwith each options contract. The annualized volatility (used in thesimulation of prices and real options valuation) would ber/t:T

t

p.

The r/(t,T) can be written in terms of the parameters of thetwo-factor model:

r2/t;T VarlnFT;t e2jTt1 e2jtr2v2j

r2n 2ejTt1 ejtqnvrvrn

j

: 9

If we again assume that the options expire at the maturity of thefutures contracts, then T t, then e2j(Tt) ej(Tt) 1. Further-more, it can be shown that as the maturity of the futures contractsincreases (T!1), the implied annualized volatility of the futurescontracts will mostly reflect the uncertainty about the long-termfactor.16 In other words, for large T, Eq. 9 simplifies to

r/T; T=T

p

r2n

q: 10

Thus, when maturities approach infinity, the implied annualvolatility approaches the volatility of long-term factor rn. We caninsert 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 rn.

Step 2: Estimation of ln and j. The diagram in Fig. 4 showsthe data points obtained from the log of futures prices with vari-ous maturity dates. We have fitted a curve (the solid line) to thediscrete data points observed in the market. From now on, thiscurve will be called the futures curve. This curve shows that thelog of the futures prices is affected by short-term volatility fornear-term maturity contracts, but as the time to maturity increases,the effect of the short-term fluctuations fades away. For long ma-turity futures, the futures curve turns into a straight line, which

has the slope of ln 1=2r2n. Having estimated rn in the previousstep, we can estimate ln, the risk-neutral drift rate for the long-term factor. This curve also reveals a rough estimation for themean-reversion coefficient j. The half-life of the deviation is thelength of time that the short-term deviations are expected to halveand is equal to ln2=j. The curve shows that the short-term

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13 We found it impossible to split the total estimate of ln into separate estimates of ln andkn, our calibration procedure directly estimates ln .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 esti-

mates of a parameter on the basis of measurements that contains noise or other inaccura-

cies. If historical spot oil prices (values for St) are considered as the measurement, then

because ln(St) nt vt, Kalman-filter methods can provide estimates of nt, which are nor-mally unobservable in the market. These estimates of nt can, in turn, be used to estimatethe parameters of the equation dnt lndt rndzn .15 Note that none of the parameter-estimation approaches will produce the correct param-

eters, 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. 16 When T t, limT1

1 ejT T

0 and limT1

1 e2jT T

0.




deviations will decrease to half its value in approximately10 months; this implies a mean-reversion coefficient ofln210=12 0:83.

Step 3: Estimation of rv and qnv. When T t and we workwith near-maturity futures contracts (T 0), it can be shown thatfor small T, Eq. 9 results in the annualized volatility as follows17:

r/T; T=T

p e2jTr2v r2nT 2ejTqnvrnrv

q: 11

Eq. 11 shows that the annualized volatility [r/t; T=T

p] for

near-maturity futures contracts (T 0) is the sum of the volatil-ities of short-term and long-term factors. Because we already cal-culated rn in Step 1, we can now insert the implied volatility ofoptions that expire into Eq. 10 (e.g., in 1 to 3 months, resulting ina system of two equations with two unknowns from which we cancalculate rv and qnv.) Any two of r/1=12; 1=12 , r/1=6; 1=6 ,or r/1=4; 1=4 , inserted into Eq. 3 will result in such a system oflinear equations.

Fig. 5 shows the implied volatility of observed options onfutures prices taken from NYMEX on 15 May 2011. The fittedvolatility curve is a helpful tool in estimating the implied volatil-ity for far-maturity options prices and extrapolating the trend tonear-maturity (T 0) options prices.18 Note that in all our analy-sis, T always t because we do not have access to other combina-tions in the market.

Step 4: Estimation of n0, v0, and kv. We can estimate n0 (thelong-term factor of the spot price at Time 0), v0 (the deviationfrom the equilibrium at Time 0), and kv (the risk premium for theshort-term factor) on the basis of the estimations performed in theprevious steps and the relationships between parameters and ini-tial state variables. In this step, the spot and futures prices provideindirect information about these unknowns. Fig. 6 depicts therelationships between the model parameters.19

The log of the current spot oil price is the summation of n0 andv0; therefore, because S0 is observed in the market [West TexasIntermediate (WTI) spot oil price was USD 99/bbl on 15 May2011], we can write v0 in terms of n0

v0 lnS0 n0: 12

Eq. 13 shows the relationship between FT,0 (price of thefutures contract at Time 0 with maturity at T) and other model pa-rameters. We have estimated ln, rn, j, rv, and qvn previously.Furthermore, on the basis of Eq. 4, we can replace v0 with ln(S0) n0 in Eq. 5. This leaves us with two unknowns, n0 and kv.

lnFT;0 ejTv0 n0 lnT 1 ejTkvj

1=2 1 e2jT r2v

2j r2nT 21 ejT

qvnrvrnj

" #:

13

. . .

. . . . . . . . . . . . . . . . . . . . . . . . . .

0

0.05

0.1

0.15

0.2

0.25

0.3

0.35

0 2 4 6 8 10

Ann

ua

l Vo

latil

ity

Time, years

Implied Volatility of Observed Option Prices

Volatility Curve

Fig. 5The implied volatility of observed options on futures inthe commodity market and the fitted volatility curve.

Time

Log (Prices)

Short-Term Risk Premium (/)

Deviation (0)

Equilibrium Price (0) 0+

2

4+

Expected Spot Prices Slope = + 2

Futures Curve Slope =

+

2

Spot Price

Fig. 6Relationships among parameters of the two-factor price process.

4.4

4.45

4.5

4.55

4.6

4.65

0 2 4 6 8 10

Log

of Pr

ices

Time, years

Observed Futures Prices

Futures Curve

Half-life = ( 2) /

Slope = * + 2

Fig. 4The futures curve and its relationship with the mean-reversion coefficient.

17 We used the fact that, when T t, limT!0

1 e2jTT

2j and limT!0

1 ejTT

j.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 cor-

rect 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 observ-

able 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.




As discussed previously, the data points corresponding to theupper curve of Fig. 6 are not observable in the market. The riskpremiums kv and kn describe the differences between the lowerand upper curve, and because expected prices are not observed,these risk premiums cannot be estimated. In the risk-neutral ver-sion of the SS model, we would need only ln (which is estimatedaccurately using the lower curve and eliminates the need forassessing kn).

The errors in the estimate of kv shift all the estimates of nt up ordown by a constant (kv=j) with vt adjusting accordingly so as topreserve the sum nt vt corresponding to the log of expected spotprice (Schwartz and Smith 2000). Assume we replace kv by kv Dfor any D, and in compensation replace v0 by v0 D j= and n0 byn0 D j= . These changes will not affect the risk-neutral stochasticprocesses. We use this property and arbitrarily set kv 0 in ourestimations (which means D kv). Then, if we use Eqs. 4 and 5to calculate v0 and n0, our estimates for these initial state variableswill be valid for the risk-neutral version of the SS model.

We can insert the logarithm of price of a futures contract intoEq. 13, and by setting kv 0, we can estimate n0 and v0 usingEqs. 12 and 13.

The estimated parameters used in our economic analysis areshown in Table 1.

The parameter-estimation procedure, the order of the esti-mates, and the relationships used are summarized in Table 2.

Project Valuation

Introduction. In the classical DCF approach to valuation, the netpresent value (NPV) of a project is calculated by discounting thefuture expected values using a discount rate that reflects the costof capital and desired rate of return. This discount rate is mark-edly higher than the prevailing risk-free interest rate and hencecan be viewed as a risk-adjusted discount rate. Furthermore, mostcorporations use a single discount rate in the analysis of all proj-ects or, at best, establish different discount rates for only a fewlarge classes (e.g., political, pipeline installation vs. new fielddevelopment) of investment decisions. This one-size-fits-allapproach to dealing with projects mimics the risks of the overallfirm, but it fails to reflect the variety of projects that feature differ-ent types and levels of uncertainty. Furthermore, using risk-

adjusted discount rates often leads to an undervaluation of numer-ous oil and gas projects with long-time horizons.

Another major criticism of the classical DCF approach is thatit is based on a static view, in which future decisions areassumed to depend only on information available now and not onadditional information that would be available when the decisionis made. This ignores management flexibility and the value that isgenerated by managements ability to make decisions during theexecution of the project. The value of this flexibility can only bedetermined using a real option valuation approach.20

Decision-tree analysis can be used to model flexibilities andoptions associated with oil and gas projects. Decisions to maxi-mize the value of the project can be included as downstream de-cision nodes, allowing the managers to respond as uncertaintiesare resolved over the projects life. The resulting decision tree canthen be solved using the same risk-adjusted discount rate judged tobe appropriate for the original project without flexibility. Unfortu-nately, although this nave approach may provide a representativemodel of the project and its management, it does not result in a cor-rect valuation of the real options associated with the project. Thisis because the optimization that occurs at each downstream deci-sion node changes the expected future cash flow of the project.This, in turn, changes the risk characteristics of the project becausethe standard deviation of the project without the flexibility is dif-ferent from that of the project with the flexibility.21

TABLE 1ANNUAL PARAMETER ESTIMATES FOR THE TWO-FACTOR OIL-PRICE MODEL

WITH ASSUMED KX50

Parameter Description Estimated Value

v0 Short-term increment of the log of the spot price 0.21n0 Long-term increment of the log of the spot price 4.38rn The volatility of the long-term factor 10%rv The volatility of the short-term factor 29%ln The risk-neutral drift rate for the long-term factor 0.5%kv The risk premium for short-term factor 0j Mean-reversion coefficient 0.83qnv The correlation coefficient between the random increments 0.3

TABLE 2SUMMARY OF PARAMETER-ESTIMATION PROCEDURE

Step 1 - Estimation

of rn

Calculate implied volatilities for a long-maturity futures contract using Eqs. 6 or 7; then use Eq. 9 and

calculate rn.Step 2 Estimation

of ln and jBuild the log of futures curve based on the observed prices of futures contracts; calculate the slope of the log

of futures curve; estimate ln by subtracting 1=2r2n from the slope.

Estimate half-life from the futures curve; use the relationship half-life ln2j

to estimate j.

Step 3 Estimation

of rv and qnv

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 rv and qnv.Step 4 Estimation

of n0, v0, and kv

Arbitrarily set kv0; use Eqs. 11 and 12 to build a system of equations; solve the system of equations andcalculate n0 and v0.

20 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 flexibil-

ity, 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 mak-

ing decisions at the face of resolving uncertainties.

21 In practice, the difficulties of evaluating any investment opportunity using a single risk-

adjusted 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 uncer-

tainties in order to apply a single risk-adjusted discount rate to all cash flows. This aggregate

picture of projects 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 sin-

gle 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 ag-

gregate risk profile of the project. Using a single risk-adjusted discount rate without consider-

ing this effect will result in evaluation biases.




To adjust the nave approach, we can use a risk-neutral pricingapproach (Smith and Nau 1995; Smith and McCardle 1999) by dis-tinguishing between market risks, which can be hedged by tradingsecurities, and private risks, which are project-specific risks. Therisk-neutral approach provides a single, coherent risk-neutralmodel, which is used to estimate the value of the project both withand without options. In this approach, the probabilities or processesassociated with the uncertainties or stochastic factors in the model(e.g., oil prices) are risk-adjusted. The value of the investment isthen calculated by determining its expected NPV using the riskadjusted probabilities or processes for market risks and true proba-bilities for private risksall discounted at the risk-free rate. Aswill be discussed in more detail in Appendix A, risks that fall some-where between market and private (e.g., rig-rate risks) are assessedas true probabilities conditional on the concurrent market state.

Combining decision-tree analysis with the risk-neutral approachprovides a consistent and correct approach to the valuation of flexi-ble projects. Unfortunately, decision trees (or binomial lattices) areawkward to use when the investment decision is a function of mul-tiple uncertainties and involves multiple decision points. The LSMapproach is well suited for such situations because it does not sufferfrom the curse of dimensionality with regard to the number ofuncertainties and decision points (Longstaff and Schwartz 2001;Willigers and Bratvold 2009). Using the LSM approach, we startby building a Monte Carlo simulation model that takes into accountall relevant uncertainties of the problem. From this model, we gen-erate a large number of possible outcomes for the project withoutoptions. In order to calculate an optimal exercise policy at each de-cision point, we need to evaluate the expected future payoff (thecontinuation value) for each possible choice, conditioned on the re-solution of all the uncertainties up to that time. The optimal policyis then achieved by selecting the option that yields the highest con-tinuation value given the information available.

The challenge is to determine the continuation value, and thisis where Longstaff and Schwartz (2001) suggest the use of linearregression. The LSM method is applied to find an approximatevalue function that relates the continuation value to the underlyinguncertainties. Once the value function has been established, theestimated continuation value is determined by the realized valuesof the uncertainties in a given period.

Abandonment-Timing Option. The timing of the decision toabandon an oil or gas field can have a significant impact on assetvalue. As a simple rule, when the revenues from selling theextracted hydrocarbons are not sufficient to cover expenses(including tax) then the field is abandoned. In reality, the uncertainoil or gas prices, complex cash-flow structures, and interrelateddecisions transform the timing of field abandonment into a com-plex real option. Most major oil and gas fields produce for 20 to 30years or more before their production rates decline below an eco-nomic limit and continuing the operations would incur a net loss tothe company. At that time, the decision of how and when to termi-nate the operations will create an important exit option. The pro-ject value is the expected sum of cash flows, including thoseassociated with any inherent options. In some projects, the aban-donment option may have a significant impact on project value.22

After the production is closed down, the installations must bedismantled and removed. Generally, in offshore installations thetopsides are taken to shore for recycling; the substructures mayalso be removed or left in place either temporarily or permanently(subject to regulations and government policies). Dismantling andremoval of large platforms is costly and involves regulatory andenvironmental considerations (Osmundsen and Tveteras 2003).The removal costs are, however, highly uncertain partly becauseof the challenging nature of the job and partly because of theindustrys lack of experience in dismantling large offshore con-

struction (Parente et al. 2006). There are more than 6,500 offshoreinstallations worldwide, and Osmundsen and Tveteras (2003) esti-mated that the cost of removal would exceed USD 20 billion.Thus, significant value loss or gain potential is associated with thetiming of the abandonment.

Given the uncertain future oil prices, the net cash flows fromoil production will also be uncertain. In this situation, the aban-donment decision must be based on the expectation of the futurecash flows, given continued production. The expected future oilprices and cash flows should be conditional on the current price,and we call the conditional expectation of the future cash flowsthe continuation value, which includes any value gained fromoptimal decisions in the future. In uncertain situations, the optimalcourse of action would be to compare the value from immediateabandonment with the continuation value and choose the alterna-tive with the largest value. This view of abandonment decisionsas an American option is discussed in Myers and Majd (1990) andBerger et al. (1996).

At the time of abandonment, the wells must be plugged andthe infrastructures dismantled. In most fiscal regimes, the operat-ing company is responsible for minimizing the risk of pollutionand keeping the production site in an environmentally acceptablestate. Performing this is costly, the resale value of the existingequipment is often insignificant, and thus the abandonment of afield generally requires a significant outlay for the operator.Osmundsen and Tveteras (2003) studied the decommissioningcosts and policy issues on the Norwegian continental shelf. Theirstudy reveals numerous sources of uncertainty that impact thecost of abandoning a field.

In some situations, the field may be acquired by another opera-tor23 or there may be a positive resale value for the facilities andinfrastructure.24 In these situations, the optimal time of abandon-ment 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, stor-age, and offloading (FPSO) vessel. At the time of the abandon-ment, the FPSO vessel can be refitted and reused for another field.Compared with a field that has been developed using fixed pro-duction platforms, the FPSO is a more-flexible asset and mayhave a positive salvage value. The salvage value may induce adifferent course of action and a different value for the abandon-ment option compared with a fixed-platform operation. Every-thing else being equal, flexible equipment that has a secondarymarket (compared with custom-built equipment that does nothave 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 ofthe project including the abandonment-timing option. Such avalue estimate may, for example, be relevant in negotiating aprice for the transfer of the operatorship to another party. The pro-duction is currently at 1 million bbl/yr and declining. This produc-tion level is achieved by spending USD 10 million/yr in the formof fixed costs and variable operating costs that, for this project,are well estimated by

average annual oil price 2=5 production rate:25 14. . .

22 Because of the long time span of numerous oil and gas projects, the abandonment deci-

sion will be made well into the future and the economic impact of this decision is not consid-

ered 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.

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 aban-

donment 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 Tveteras (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 simpli-

fying 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).




If we choose to continue the production in the next year, theUSD 1 million will be expensed irrespective of the amount of pro-duction. However, the operating cost varies according to the levelof production and includes the cost of pumping the liquids to thesurface, primary treatments, separation, and transporting the oil tothe supply point. The field is in late life, and the annual produc-tion decline has for several years been stable at 10% per year. His-torically, a good estimate of the variable cost in each year hasbeen given.

For the purpose of this paper, we combine the decommission-ing costs and salvage values into a single uncertain variable,which we call the abandonment cash flow, ht, which includes alldecommissioning costs and the resale value of the FPSO and thusmay take on positive or negative values. Assume the example oilfield discussed earlier is producing through an FPSO that has apositive value in a secondary market. The operator expects thatthe abandonment cash flow, ht, of the project is time dependentand can be modeled by an OU process as follows:

dh ah hdt rhdzh; 15

where a is the mean-reversion rate and rh represents the volatilityof the abandonment cash flow in which the entire abandonmentcash flow is assumed to occur in the year the field is abandoned.26

The operator realizes that the salvage value of the FPSO is corre-lated with the oil prices, while the decommissioning costs areassumed to be uncorrelated with the prices. Therefore, ht includesrisks that fall somewhere between the notions of private and mar-ket risks. Willigers (2009) looked at historical rig data and foundthe correlation between oil price and rig rental rates to be approxi-mately 0.8. Because only a fraction of ht is correlated with the oilprices,27 the operator decides to use a correlation coefficient of0.2 for this uncertainty (i.e., qSh 0.2). The other parameters ofthe OU process are shown in Table 3.

The LSMMethod. Although the literature recognizes the impor-tance of uncertainty in abandonment costs and salvage values(Myers and Majd 1990), valuing flexibility in light of multipleuncertainties is difficult using traditional option valuation tools.With two uncertainties (state variables), one or two trinomial treesare required and any stochastic process beyond GBM inducestimestep-dependent up and down probabilities (Hahn and Dyer2008). Going beyond two state variables generally requires someform of simulation. In this subsection, we illustrate how the LSMmethod (Longstaff and Schwartz 2001) can incorporate the keyuncertainties (future oil prices, which themselves consists of twounderlying state variables, and abandonment cash flows) into theabandonment-timing option, where the uncertainties are treated ina more-realistic manner than by using GBM processes.

The LSM approach to option valuation starts by building aMonte Carlo simulation model that takes into account all relevantuncertainties of the problem. Using this model, we generate a

large number of possible outcomes for the continue to producesituation. In order to calculate the optimal abandonment policy ateach decision point, we need to evaluate the expected future pay-off (the continuation value) for each possible choice (abandon orcontinue production), conditioned on the resolution of all theuncertainties (oil price and abandonment cash flow) up to thattime. The optimal policy is then achieved by selecting the optionthat yields the highest value given the information available.

The challenge is to determine the continuation value, and thisis where Longstaff and Schwartz (2001) suggested the use of lin-ear regression. The LSM method is applied to find an approxima-tion to the conditional value expectation given the realized, orcurrent, values of the uncertainties in a given time period.

In this work, we want to assess the value of the flexibility toabandon the field at any given point in time. The optimal aban-donment time will be assessed on the basis of the information wehave about the production level, decline rate, oil-price dynamics,and abandonment-cash-flow fluctuations. In our analysis, the cashflows before Year 0 are treated as sunk costs and will not affectany calculations. We assume the field must be abandoned withinthe next 15 years.28 The cash flow at any given year depends onthe (average) oil price on that year, St; the abandonment cash flowof the project, ht; and the decision, dt, to abandon or to continuethe project. We denote the payoff in the period (t, t Dt) as P(St,ht, dt, t). We denote the present value of all future cash flows (onthe basis of optimal decisions and discounted at the risk-free dis-count rate, r) as F(St, ht, t). The optimal decision at Year t is tochoose the course of action that maximizes the present value ofcurrent cash flows along with the expected cash flows from opti-mal decisions in the future. This is shown as the followingdynamic programming equation:29

FSt; ht; t maxd

PSt; ht; dt; t erDtEFStDt; htDt; t Dt

; 16

where E() is the expectation operator. We use the integrated risk-neutral valuation procedure described in Smith and Nau (1995).The oil price is treated as a market uncertainty, which can behedged in the commodity markets using suitable market instru-ments. The abandonment cash flow is a technical uncertainty, butobviously has a correlation with oil prices. In our model, wedetermine the expected yearly cash flows by calculating trajecto-ries for oil prices and abandonment cash flows using the risk-neu-tral processes. All discounting is performed at a risk-free rate of3%. See Appendix A for further discussion on risk-neutralvaluation.

Because we will use Monte Carlo simulation to generate pricesand cash flows, we need to discretize the SS processes and themean-reverting process that describes the abandonment cash flow.The following time-discrete equations have been used.

lnst vt nt : 17

. . . . . . . . . . . . . . . . . . . .

. . . . . . .

. . . . . . . . . . . . . . . . . . . . . . . . . .

TABLE 3YEARLY PARAMETERS FOR THE MEAN-REVERTING PROCESS DESCRIBING

THE DYNAMICS OF THE ABANDONMENT CASH FLOWS

Description Parameter Value

Mean-Reversion Coefficient a 1Mean Abandonment Cash Flow h USD 30 Million

Volatility of Abandonment Cash Flows rh USD 9 million/yr (30% annual volatility)

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 inthe 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 adjust-

ing this uncertainty. In general, if the private uncertainties are correlated with market uncer-

tainty, 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 prob-

lem. 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 opti-

mality associated with our dynamic programming model. See Bertsekas (2005) for an intro-

duction to dynamic programming.




The discretization for the long-term component, Eq. 5, is

nt nt1 lnDt rnenDt

p: 18

Eq. 4 is the continuous time version of the stationary first-order autoregressive process in discrete time (Dixit and Pindyck1994). The exact discrete model (valid for large Dt) for the risk-neutral short-term component process is

vt vt1ejDt 1 ejDtkvj rvev

1 e2jDt

2j

r;

19where en and ev in Eqs. 18 and 19 are standard normal randomvariables and are correlated in each time period with the correla-tion coefficient qnv. It can be shown (Wiersema 2008) that if enand e are two independent normal random variables, we can corre-late the price processes in Eqs. 18 and 19 by constructing the ran-dom variable for the second process as

ev enqnv e1 q2nv

q: 20

The risk-neutral process used for simulation of abandonmentcash flows is

ht ht1eaDt h1 eaDt rheh1 e2aDt

2a

r: 21

Again eh is the standard normal random variable. As we dis-cussed in the preceding subsection, the uncertainty about theabandonment cash flow is technical uncertainty but has a correla-tion with the oil prices. We used Eq. 20 to correlate the abandon-ment cash flow with the long-term factor of the oil prices.

We can solve this dynamic programming problem by startingat the final date T Year 15 and then working backward to thecurrent date by solving the associated Bellman equation at all thedecision dates t Year 15, , Year 1. Our decisions should takeinto account the expectation of the uncertainties about future time

periods. It is difficult to directly assess the conditional expectedcontinuation value E[F(StDt, VtDt, t Dt)], and, as discussednext, we approximate this by a regression function. In this paper,we numerically solve the dynamic-programming model using theLSM method.

The LSM algorithm uses the regression equation as the estima-tor of the future values conditional on the current information (theconditional expectation value). In this regression, the current statevariables (simulated oil prices and abandonment cash flows) arethe independent variables. The decision outcomes (cash flowsreceived in the future) are then used to predict the dependent vari-able. With this regression equation, one can simply insert currentstate values and obtain a prediction for the future values.

We ran a simulation of 10,000 trials and determined the near-optimal decisions in each year (the details of the LSM implemen-tation are discussed in Appendix B). Using these near-optimaldecision policies, we found an expected NPV of USD 107.92 mil-lion for the project.

Analysis of Results and Discussion. The option value calculatedin the preceding subsection is an expected value and is based onour modeling of the underlying uncertainties. While this numbergives a positive option-value estimate, it may be useful to alsolook at the range of possible outcomes and associated frequencies.In Fig. 7, the output of the LSM simulation has been presented inthe form of a frequency diagram. This figure shows that on the ba-sis of the results of this simulation, some scenarios result in NPVvalues higher or lower than USD 107.92 million.

It would also be beneficial to know what the optimal decisionis on the basis of the value of the state variables. For example, ifin Year 2 the oil price is S2 and the abandonment cash flow is h2,then what would be the optimal decision? For what range of thestate variables should the decision maker abandon the project andfor what range should the operations continue? Fig. 8 shows apolicy map that can provide insight for these questions.

Fig. 9 shows the outputs of the LSM algorithm presented in adifferent format. This diagram shows the probability that the pro-ject is abandoned in each year, given the information available inYear 0. Such presentations can provide valuable insight for thedecision makers. As an example, this diagram reveals that in5,519 iterations out of 10,000 iterations, the optimal decision wasto abandon the project in the first year.30

We can compare the value of the abandonment-timing optionwith the case in which the field has to be abandoned no later thana particular time (when, for example, contracts or authorities forcethe company effectively to abandon the field no later than a par-ticular time). Fig. 10 shows the market value of the project for arange of different levels of operating flexibility. The horizontalaxis shows the latest possible time for the project to be aban-doned. For example Year 2 means the project can be abandonedin either Year 1 or Year 2.

. . . . . . . . . . . . . . . . .

. . . . . . . . . . . . . . . . . . . . .

. . . .

0

500

1,000

1,500

2,000

2,500

34.0

845

.22

56.3

767

.52

78.6

789

.81

100.

9611

2.11

123.

2513

4.40

145.

5515

6.70

167.

8417

8.99

190.

1420

1.28

212.

4322

3.58

234.

7224

5.87

257.

02

Frequ

ency

Project value, USD Million

Fig. 7Distribution of the project outcomes.

Second Year Oil Price, S2 (USD/bbl)

Aba

ndo

nmen

t Cas

h Fl

ow

, 2

,

USD

Milli

on

15 60 110 160 210 260

1

1

7

3

Abandon the Oil Field

Continue Operation

Fig. 8Decision map for Year 2.

0.55

190.

2033

0.10

20.

0548

0.03

260.

0179

0.01

230.

0088

0.00

570.

003

0.00

210.

0024

0.00

130.

0007

0.00

12

0

0.1

0.2

0.3

0.4

0.5

0.6

Pro

babi

lity o

f Aba

ndo

nm

en

t

Fig. 9Probability of abandoning the project in each year,given the information available at Year 0.

30 Note that Fig. 9 shows the probabilities for abandoning the field conditional on the infor-

mation in Year 0. The decision maker may also want to know the probabilities of abandon-

ing 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.




Clearly, from Fig. 10, increased flexibility with regard to theabandonment time can add significant value. For this example, thevalue increase is largest in the first years and then levels off atapproximately Year 7.

Conclusions

In this paper, we have illustrated the details of the parameteriza-tion, using the implied volatility approach with futures andoptions data, and implementation of the SS two-parameter sto-chastic price process. We then implemented a real option valua-tion model for an abandonment option using the LSM formulationand the SS price process.

The real options paradigm recognizes the value-creation poten-tial resulting from decision makers active management of theirinvestments over time. Modeling the price uncertainty using atwo-factor stochastic price process provides a realistic implemen-tation of oil-price dynamics and, thus, a realistic option value.

The LSM method implemented here readily generalizes tomore-complex problems. We used a second-degree polynomialequation in our regression to estimate the conditional expectedvalue of continuation in our example. There are other problems inwhich using a simple regression function is unsuitable [see Sten-toft (2004) for discussion on the convergence of this estimator tothe true conditional expectation value]. In general, we can addany number of terms, including nonlinear terms to the regressionequation as required by the problem context. The complexity ofthe regression equation does not impose limitations on the effi-ciency of the LSM algorithm, and the LSM algorithm is relativelyinsensitive to the number of uncertainties in the problem, but itscomplexity grows with the number of decision points and thenumber of alternatives at each decision point.

We also illustrated an implied approach for calibration of theSS model to market data. This method of parameter estimation isrelatively simple and practical, relies on forward market data asopposed to historical data, and should encourage analysts and de-cision makers to include realistic models of oil-price uncertaintiesin their valuation efforts. It should be noted that there is uncer-tainty or variability in the model parameters because the calibra-tion is based on futures prices that change on a frequent basis. Ifthis is a concern, it is possible to use average values or valuesdetermined by the decision maker (e.g., the chief financial officerfor numerous oil companies). For a better understanding of theimpact of this uncertainty, a sensitivity analysis of project valuesand associated decisions based on the variability of model param-eters should always be conducted.

In this paper, we analyzed a generic example of project aban-donment without considering any tax effect. Clearly, analyzingthe abandonment option under different tax regimes may have asignificant impact on the optimal abandonment time.

Acknowledgments

We thank James E. Smith for his response to our questions. Wealso thank the two anonymous reviewers and Associate Editor

Steve Begg, whose valuable comments, suggestions, and insightshelped improve this paper.

References

Baker, M.P., Mayfield, E.S., and Parsons, J.E. 1998. Alternative Models

of Uncertainty Commodity Prices for Use with Modern Asset Pricing

Methods. Energy Journal 19 (1): 115148. http://dx.doi.org/10.5547/ISSN0195-6574-EJ-Vol19-No1-5.

Begg, S.H. and Smit, N. 2007. Sensitivity of Project Economics to Uncer-

tainty in Type and Parameters of Price Models. Paper SPE 110812 pre-

sented at the SPE Annual Technical Conference and Exhibition,

Anaheim, California, USA, 1114 November. http://dx.doi.org/10.2118/

110812-MS.

Berger, P.G., Ofek, E., and Swary, I. 1996. Investor valuation of the aban-

donment option. Journal of Financial Economics 42 (2): 257287.

http://dx.doi.org/10.1016/0304-405x(96)00877-x.

Bertsekas, D.P. 2005. Dynamic Programming and Optimal Control, thirdedition, Vol. 1. Belmont, Massachusetts: Athena Scientific.

Bickel, J.E. and Bratvold, R.B. 2008. From Uncertainty Quantification

to Decision Making in the Oil and Gas Industry. Energy Explora-

tion and Exploitation 26 (5): 311325. http://dx.doi.org/10.1260/014459808787945344.

Brennan, M.J. and Schwartz, E.S. 1985. Evaluating Natural Resource

Investments. The Journal of Business 58 (2): 135157.Carriere, J.F. 1996. Valuation of the early-exercise price for options using

simulations and nonparametric regression. Insurance: Mathematics

and Economics 19 (1): 1930. http://dx.doi.org/10.1016/s0167-6687(96)00004-2.

Cortazar, G. and Schwartz, E.S. 1994. The Valuation of Commodity Con-

tingent Claims. The Journal of Derivatives 1 (4): 2739. http://dx.doi.org/10.3905/jod.1994.407896.

Cortazar, G. and Schwartz, E.S. 2003. Implementing a stochastic model

for oil futures prices. Energy Economics 25 (3): 215238. http://

dx.doi.org/10.1016/S0140-9883(02)00096-8.

Cox, J.C., Ingersoll, J.E. Jr., and Ross, S.A. 1985. An Intertemporal

General Equilibrium Model of Asset Prices. Econometrica 53 (3):

363384. http://dx.doi.org/10.2307/1911241.

Cox, J.C. and Ross, S.A. 1976. The valuation of options for alternative sto-

chastic processes. Journal of Financial Economics 3 (12): 145166.

http://dx.doi.org/10.1016/0304-405x(76)90023-4.

Cox, J.C., Ross, S.A., and Rubinstein, M. 1979. Option pricing: A simpli-

fied approach. Journal of Financial Economics 7 (3): 229263. http://

dx.doi.org/10.1016/0304-405x(79)90015-1.

Dias, M.A.G. 2004. Valuation of exploration and production assets: an

overview of real options models. J. Pet. Sci. Eng. 44 (12): 93114.

http://dx.doi.org/10.1016/j.petrol.2004.02.008.

Dias, M.A.G. and Rocha, K.M.C. 1999. Petroleum Concessions with

Extendible Options Using Mean Reversion with Jumps to Model the

Oil Prices. Paper prepared for presentation at the 3rd International

Conference on Real Options, Wassenaar/Leiden, The Netherlands, 68

June. http://www.realoptions.org/papers1999/MarcoKatia.pdf.

Dickey, D.A. and Fuller, W.A. 1981. Likelihood Ratio Statistics for

Autoregressive Time Series with a Unit Root. Econometrica 49 (4):

10571072. http://dx.doi.org/10.2307/1912517.

Dixit, A.K. and Pindyck, R.S. 1994. Investment Under Uncertainty.Princeton, New Jersey: Princeton University Press.

Geman, H. 2000. Scarcity and price volatility in oil markets. Technical

Report, EDF Trading, London.

Geman, H. 2005. Commodities and Commodity Derivatives: Modellingand Pricing for Agriculturals, Metals and Energy. West Sussex, UK:

John Wiley & Sons.

Gibson, R. and Schwartz, E. 1990. Stochastic Convenience Yield and the

Pricing of Oil Contingent Claims. The Journal of Finance 45 (3):

959976. http://dx.doi.org/10.2307/2328801.

Glasserman, P. 2004. Monte Carlo Methods in Financial Engineering,Vol. 53. New York: Applications of MathematicsStochastic Model-

ling and Applied Probability, Springer-Verlag.

Hahn, W.J. and Dyer, J.S. 2008. Discrete time modeling of mean-reverting

stochastic processes for real option valuation. European Journal of

Operational Research 184 (2): 534548. http://dx.doi.org/10.1016/j.ejor.2006.11.015.

81.2

6

91.9

8 97.8

4

101.

75

104.

02

105.

43

106.

41

106.

96

107.

31

107.

51

107.

66

107.

79

107.

87

107.

92

107.

92

707580859095

100105110

Proj

ect V

alue

, USD

Mill

ion

Degree of Flexibility

Fig. 10Project value for different ranges of operatingflexibilities.




Harrison, J.M. and Pliska, S.R. 1981. Martingales and stochastic integrals in

the theory of continuous trading. Stochastic Processes and their Applica-tions 11 (3): 215260. http://dx.doi.org/10.1016/0304-4149(81)90026-0.

Harvey, A.C. 1989. Forecasting, Structural Time Series Models and theKalman Filter. Cambridge, UK: Cambridge University PressCam-

bridge University Press.

Hilliard, J.E. and Reis, J. 1998. Valuation of Commodity Futures and

Options Under Stochastic Convenience Yields, Interest Rates, and

Jump Diffusions in the Spot. The Journal of Financial and Quantita-tive Analysis 33 (1): 6186. http://dx.doi.org/10.2307/2331378.

Kalman, R.E. 1960. A New Approach to Linear Filtering and Prediction

Problems. Journal of Basic Engineering 82 (1): 3545. http://dx.doi.org/10.1115/1.3662552.

Kulatilaka, N. 1995. 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, 89108. Westport, Con-

necticut: Praeger Publishers.

Laughton, D.G. and Jacoby, H.D. 1993. Reversion, Timing Options, and

Long-Term Decision-Making. Financial Management 22 (3):225240.

Laughton, D.G. and Jacoby, H.D. 1995. The Effects of Reversion on Com-

modity Projects of Different Length. In Real Options in Capital Invest-ment: Models, Strategies, and Applications, third printing edition, ed.L. Trigeorgis, Part IV, 185206. Westport, Connecticut: Praeger

Publishers.

Longstaff, F.A. and Schwartz, E.S. 2001. Valuing American options by

simulation: A simple least-squares approach. Rev. Financ. Stud. 14(1): 113147. http://dx.doi.org/10.1093/rfs/14.1.113.

Luenberger, D.G. 1998. Investment Science. New York: Oxford University

Press.

Merton, R.C. 1973. An Intertemporal Capital Asset Pricing Model. Econo-metrica 41 (5): 867887. http://dx.doi.org/10.2307/1913811.

Myers, S.C. and Majd, S. 1990. Abandonment Value and Project Life.

Advances in Futures and Options Research 4: 121.

Osmundsen, P. and Tveteras, R. 2003. Decommissioning of petroleum

installationsmajor policy issues. Energy Policy 31 (15): 15791588.http://dx.doi.org/10.1016/s0301-4215(02)00224-0.

Paddock, J.L., Siegel, D.R., and Smith, J.L. 1988. Option Valuation of

Claims on Real Assets: The Case of Offshore Petroleum Leases. TheQuarterly Journal of Economics 103 (3): 479508. http://dx.doi.org/

10.2307/1885541.

Parente, V., Ferreira, D., Moutinho dos Santos, E. et al. 2006. Offshore

decommissioning issues: Deductibility and transferability. Energy Pol-icy 34 (15): 19922001. http://dx.doi.org/10.1016/j.enpol.2005.02.008.

Pilipovic, D. 1998. Energy Risk: Valuing and Managing Energy Deriva-

tives. New York: McGraw-Hill.

Pindyck, R.S. 1999. The Long-Run Evolution of Energy Prices. TheEnergy Journal 20 (2): 127.

Pindyck, R.S. 2001. The Dynamics of Commodity Spot and Futures Mar-

kets: A Primer. The Energy Journal 22 (3).

Schwartz, E.S. 1994. Review of Investment Under Uncertainty, Dixit A.K.

and Pindyck R.S. The Journal of Finance 49 (5): 19241928.

Schwartz, E.S. 1997. The Stochastic Behavior of Commodity Prices:

Implications for Valuation and Hedging. The Journal of Finance 52(3): 923973. http://dx.doi.org/10.1111/j.1540-6261.1997.tb02721.x.

Schwartz, E. and Smith, J.E. 2000. Short-Term Variations and Long-Term

Dynamics in Commodity Prices. Management Science 46 (7):893911. http://dx.doi.org/10.1287/mnsc.46.7.893.12034.

Smith, J.E. and McCardle, K.F. 1999. Options in the Real World: Lessons

Learned in Evaluating Oil and Gas Investments. Oper. Res. 47 (1):115. http://dx.doi.org/10.1287/opre.47.1.1.

Smith, J.E. and Nau, R.F. 1995. Valuing Risky Projects: Option Pricing

Theory and Decision Analysis. Management Science 41 (5): 795816.http://dx.doi.org/10.1287/mnsc.41.5.795.

Smith, J.E. 2005. Alternative Approaches for Solving Real Options Prob-

lems (Comment on Brandao et al. 2005). Decision Analysis 2 (2):

89102. http://dx.doi.org/10.1287/deca.1050.0041.

Stentoft, L. 2004. Convergence of Least Squares Monte Carlo Approach

to American Option Valuation.Management Science 50 (9).

Tsitsiklis, J.N. and van Roy, B. 1999. Optimal stopping of Markov

processes: Hilbert space theory, approximation algorithms, and an

application to pricing high-dimensional financial derivatives IEEETransactions on Automatic Control 44 (10): 18401851. http://dx.doi.org/10.1109/9.793723.

Tsitsiklis, J.N. and van Roy, B. 2001. Regression methods for pricing

complex American-style options IEEE Trans. Neural Networks 12 (4):694703. http://dx.doi.org/10.1109/72.935083.

Uhlenbeck, G.E. and Ornstein, L.S. 1930. On the Theory of the Brownian

Motion. Physical Review 36 (5): 823841. http://dx.doi.org/10.1103/

PhysRev.36.823.

West, M. and Harrison, J. 1996. Bayesian Forecasting and Dynamic Mod-

els, second edition. New York: Springer-Verlag.

Wiersema, U.F. 2008. Brownian Motion Calculus. West Sussex, UK: John

Wiley & Sons.

Willigers, B.J.A. 2009. 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 124180 presented at the SPE Annual

Technical Conference and Exhibition, New Orleans, 47 October.

http://dx.doi.org/10.2118/124180-MS.

Willigers, B.J.A. and Bratvold, R.B. 2009. Valuing Oil and Gas Options

by Least-Squares Monte Carlo Simulation. SPE Proj Fac & Const 4(4): 146155. SPE-116026-PA. http://dx.doi.org/10.2118/116026-PA.

Appendix A: Risk-Neutral Valuation ofUncertainties That Fall Between Marketand Private Uncertainties

Traditional DCF analysis considers uncertainty in cash flows as asource of risk. In order to reflect the effects of risk on the projectvalue, the common approach has been to risk adjust the discountrate. In performing this, we make a number of assumptions. First,when we risk adjust the discount rate, we treat all cash flows (andelements of cash flows subject to various sources of risk) equally,as if the risk is distributed uniformly across all elements of thecash flows.31 Second, using a single risk-adjusted discount ratefor valuations also implies a constant risk level over time, whereasthe risks of a project usually vary with time.32 Third, when therisk-adjusted discount rates are used in decision trees or dynamicprograms, at each decision point the risk characteristics of a pro-ject will change as a result of the optimization. This will result inan underestimation of project value.

In the risk-neutral valuation technique (Cox et al. 1985),33 thevalue of an investment in a complete market is the discountedexpected value of its future payoffs under the risk-neutral proba-bilities. In other words, we risk adjust probabilities and stochasticprocesses associated with uncertainties in the model. Then, wecalculate the expected NPV of the investment using these risk-neutral probabilities or processes and discount them at the risk-free rate. On the basis of the equilibrium risk-neutral schemedescribed in Cox et al. (1985), the risk premium ki for uncertainVariable i on the basis of Mertons (1973) intertemporal capitalasset pricing model is a function of rim (the covariance betweenthe uncertain factor and the market), r2m (the variance of the mar-ket), r (the risk-free rate), and rm (the expected return on the mar-ket portfolio), following the equation

ki rimr2mrm r: A-1

In Eq. A-1,rimr2m

can be regarded as beta of the uncertain Vari-

able i. Such a risk premium can be used, for example, to riskadjust 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 projects 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 discrete-

time models and Harrison and Pliska (1981) for continuous models.




the process. If an uncertain factor is uncorrelated to the marketmovements, then its risk premium is zero and does not need riskadjustment. All the cash flows generated using these probabilitiesand stochastic processes are discounted using the risk-free dis-count rate [see Luenberger (1998) for a discussion on risk-neutralvaluation].

Smith and Nau (1995) proposed a modification to thisapproach in which uncertainties are divided into market uncer-tainties (those that can be hedged in the market by trading secur-ities) and private uncertainties (those that cannot be hedged in themarket). The expected NPV of the investment is calculated usingthe risk-adjusted probabilities or processes for market uncertain-ties and assessed probabilities for private uncertainties. Thedependence between market and private risks is captured byassessing true probabilities for the private risks conditional on thecontemporaneous market state (Smith 2005).

We may not always be able to categorize all uncertainties intoeither market or private uncertainties. Some uncertainties(e.g., the uncertainty about abandonment cash flow in our exam-ple) may fall somewhere between these two categories. However,we should note that only those uncertainties that can be hedgedwith market instruments are considered market uncertainties.Oil-price uncertainty is an example of a market uncertainty be-cause in major commodity markets there are futures and optionson futures contracts available to hedge the oil-price uncertainty.34

The price of other crudes produced from specific fields should bejudged on the basis of their correlation with traded crude types andmay not be regarded easily as market uncertainty. Those uncertain-ties that cannot be hedged using market instruments should beregarded as private uncertainties, even though they may be relatedto market uncertainties. As discussed by Smith (2005), if privateuncertainty p has a correlation with market uncertainty m, then inthe decision-analytic risk-neutral approach, we should use the trueprobabilities for p conditional on the contemporaneous state of m.If we assume p and m are correlated with correlation coefficientqpm, then the conditional stochastic process for pwill have a reduc-tion equal to bp(lm r), where r is the risk-free rate, lm is thereturn on market uncertainty m, and bp qpm

rprm

rpmr2m

can be

interpreted as beta of uncertainty p in a fashion similar to Eq. A-1.We expect that the risk-neutral valuation scheme is a more-

consistent valuation approach in that each individual uncertainty is

risk adjusted separately, whereas risk adjusting the discount ratefor the aggregate project level, although possible, is prone to errorsand requires harder judgments. Once the uncertainties are riskadjusted, they also can be used across different projects. Further-more, it is more plausible to assume that risk adjustments foruncertainties stay constant over time. Finally, projects with optionscan be modeled more consistently using this method becausechanging the course of a project can alter exposure to some uncer-tainties but will not affect the risk characteristic of the individualfactors.

Appendix B: Numerical Valuation

A simulation with 10,000 trials generates 10,000 paths for oil pri-ces and abandonment cash flows from the equations offered inThe LSM Method section. These values are then used in a spread-sheet to create 10,000 paths for the Year 0 to Year 15 cashflows.35 Out of all simulated paths, seven paths are presented inTables B-1 and B-2. These paths represent seven scenarios thatwill result in different abandonment decisions. For example, inPath 1, the cash-flow becomes negative in Year 7 because of adecrease in the oil prices and decline in the production rate. Theyearly cash flow will change sign two more times before the pro-ject is abandoned in Year 14.

The owner may abandon the project any time between Years 1and 15. If the owner decides to abandon, he or she will receivethat years cash flow and the prevailing abandonment cash flow.In some other cases, it may be better to continue the productionan

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