SPE 160000

A Two-Factor Price Process for Modeling Uncertainty in the Oil Prices Babak Jafarizadeh, Statoil ASA Reidar B. Bratvold, University of Stavanger

Copyright 2012, Society of Petroleum Engineers This paper was prepared for presentation at the SPE Hydrocarbon, Economics, and Evaluation Symposium held in Calgary, Alberta, Canada, 24–25 September 2012. This paper was selected for presentation by an SPE program committee following review of information contained in an abstract submitted by the author(s). Contents of the paper have not been reviewed by the Society of Petroleum Engineers and are subject to correction by the author(s). The material does not necessarily reflect any position of the Society of Petroleum Engineers, its officers, or members. Electronic reproduction, distribution, or storage of any part of this paper without the written consent of the Society of Petroleum Engineers is prohibited. Permission to reproduce in print is restricted to an abstract of not more than 300 words; illustrations may not be copied. The abstract must contain conspicuous acknowledgment of SPE copyright.

Abstract The petroleum industry has recognized that a consistent probabilistic approach provides improved understanding and insights into the investment decisions. Yet, although most oil & 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. We illustrate the implementation and calibration of the two-factor stochastic price model (Schwartz and Smith, 2000) that allows mean-reversion in short-term price deviations and uncertainty in the long-term equilibrium level. It provides advantages over more basic methods but is still simple enough to be communicated to corporate decision makers. The balance between realism and ease of communication of the model has led us to choose this model in favor of one-factor models, which assume that only one source of uncertainty contributes to the uncertainty in prices, or other multi-factor models where two or more factors contributes to the uncertainty in prices. Previously, a Kalman filter was used to estimate the model parameters based on historical spot and futures prices. We illustrate how current market information (such as futures prices and options on futures observed in commodity futures exchanges) can be utilized to assess the parameters of the two-factor price model. As opposed to the Kalman filter technique, the implied approach to parameter estimation is easy and intuitive, and it will generate estimates that are good enough for most valuation assessments. Introduction 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 go bad. 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. Many oil & 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 & gas investment opportunities. Valuing these flexibilities requires us to ask and answer some questions we usually do not address in traditional decision analysis. In decision tree 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 walk2; i.e., to consider them as stochastic processes that evolve over time (some references are Laughton and Jacoby, 1993, 1995; Cortazar and Schwartz, 1994; Dixit and Pindyck, 1994; Pilipovic, 1998; Schwartz, 1997; Schwartz and Smith, 2000; Geman, 2005).

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. 2 The fact that commodity prices are unpredictable creates a need for price modeling. In this paper we do not provide forecasting methods, as 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.

2 SPE 160000

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, !!, 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 model3. 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 multi-factor models where two or more factors contributes to the uncertainty in prices (see for example Schwartz, 1997, Geman, 2000, or 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 based on 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 utilized 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 easy and intuitive, and it will generate estimates that are good enough for most valuation assessments. This paper contributes to the SPE literature by: (1) familiarizing the readers 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 an investment problems. The next section introduces relevant stochastic price processes and reviews some key literature. We then introduce the SS model in section 3 and illustrate the mechanics on the implied volatility parameter estimation approach. In section 4 we discuss some challenges and conclude. 2 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 & 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 just 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 derivative4 trading horizons up to 10 years or more. These derivative 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 very 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 for example Brennan and Schwartz, 1985, or Paddock, Siegel, and Smith, 1988 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 famous 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 towards the equilibrium 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 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 and 2001).5 The 3 Two-factor stochastic price models have also been discussed in other works such as Pilipovic (1998) and Baker et al. (1998). Pindyck (1999) argues that the oil prices should be modeled using a stochastic model that reverts towards a stochastically fluctuating trend line. 4 A derivative can be defined as a financial instrument whose value depends upon (or derives from) the value of other basic underlying variables. Very often, the variables underlying derivatives are the prices of traded assets. For example, oil price futures, forwards, or 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 prices. 5 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 many years of data to determine with any degree of confidence whether a variable like oil price is mean-reverting. For example, using oil price time-series of about 30 years fails to reject the GBM hypothesis. Pindyck (1999) rejects the GBM hypothesis only after considering more than 100 years of oil price data.

SPE 160000 3

mean-reverting price process has been used for price modeling6 in a number of oil & gas related studies (examples are Laughton and Jacoby, 1993, 1995; Cortazar and Schwartz, 1994; Dixit and Pindyck, 1994; Smith and McCardle, 1999; Dias, 2004; and in the SPE literature Begg and Smit, 2007; Willigers and Bratvold, 2009). The effect of modeling a price process that is actually mean reverting with a Geometric Brownian Motion can be a significant overestimation of uncertainty in the resultant cash flows. This, in turn, can result in overstated option values. Figure 1 shows a comparison of GBM and a mean reversion process with the same volatility.

Figure 1 - Comparison of Geometric Brownian Motion (GBM) and Ornstein-Uhlenbeck mean-reverting (OU) price processes

As noted by several authors (Dias and Rocha, 1998; Dias, 2004; Geman, 2005; Begg and Smit, 2007), a key characteristic of oil prices is that their volatility appears to consist of “normal” fluctuations plus 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 mean-reverting process with a Poisson process with the additional assumption that the two processes are independent.

Although the one-factor model7 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 (1997), 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 explicitly model uncertainty 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,8 or in the risk-free interest rate. Pindyck (1999) argues that the actual behavior of real prices over the past century implies that the oil price models should incorporate mean-reversion to a stochastically fluctuating 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 that include mean-reversion. The first of these three models was a simple one-factor model where the logarithm of the price is assumed to follow an Ornstein-Uhlenbeck (OU) (Uhlenbeck and Ornstein, 1930) process. The second and third models were two-factor models. Schwartz 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).

3 The Schwartz and Smith 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 reverts.9 The equilibrium prices are modeled as a Brownian motion, reflecting expectations of the exhaustion of 6 Uhlenbeck and Ornstein (1930) introduced the first mean-reverting model. This model has been applied in biology physics, and recently in finance, commodity derivatives pricing, and petroleum valuation to describe the tendency of a measurement to return towards a mean level. 7 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 as 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. 8 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. 9 The convenience yield is implicit in the SS model as opposed to the Gibson and Schwartz model where it is modeled explicitly.

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4 SPE 160000

existing supply, improved exploration and production technology, inflation, as well as 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. 3.1 The Schwartz and Smith (SS) Model Let!!! be the commodity price at time!!, then

!" !! ! !! ! !!

(1)

where!!!! is the long-term equilibrium price level and!!! is the short-term deviation from the equilibrium prices. The long-term factor is modeled as a Brownian motion with drift rate!!! and volatility!!! .

!"! ! !!!" ! !!!"!

(2)

The short-term factor is modeled with a mean-reverting process with mean-reversion coefficient!!10 and volatility!!!. !"! ! !!!!!" ! !!!"!

(3)

where !"! and!!"! are correlated increments of standard Brownian motion processes with !!"!!"! ! !!"!". As shown in Appendix C, the model includes the Geometric Brownian Motion and Geometric Ornstein-Uhlenbeck models as special cases when there’s only uncertainty in either the long-term or short-term prices, respectively. Figure 2 shows the P10 and P90 “confidence bands” as well as the expected values for oil prices generated by the two-factor process conditional on an initial price of $99 per bbl and market information observed on May 15th 2011. These confidence bands show that at a specific time there is a 10% and 90% chance (respectively) that the prices fall below that amount.

Figure 2 - Confidence bands for the real stochastic process used for modeling the oil prices Figure 2 shows that the expected spot and equilibrium prices will be equal after a few years. This is due to 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 due to 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 much 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 two-factor 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). 3.2 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 Section 4) and will thus need the risk-neutral process to describe the dynamics of the oil prices. The implied parameter estimation 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 as the following equations

!"!! ! !!!! ! !! !" ! !!!"!!

(4)

10 The mean-reversion coefficient!!, describes the rate at which the short-term deviations are expected to disappear. Using!!, we can calculate the “half-life” of the deviations,!!"!!!! ! which is the time in which a deviation in!! is expected to halve.

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SPE 160000 5

!"!! ! !! ! !! !" ! !!!"!!

(5)

where, as before,!!"!! and!!"!! are correlated increments of the standard Brownian motion such that!!"!!!"!! ! !!"!" and where!!! and!!! are risk premiums which are being subtracted from the drifts of each process.11 The risk-neutral short-term factor is now reverting to!!!!! ! instead of zero as in the real process. The drift of the long-term factor in this model is !!! ! !! ! !! . Figure 3 compares the P10/P90 confidence bands of the spot prices and the risk-neutral prices.

Figure 3 - Confidence bands for the spot and risk-neutral price processes 3.3 Model Calibration In the SS model the commodity prices are mean-reverting towards a stochastically fluctuating equilibrium level. This model has a total of seven parameters !!! !!!! !!! ! !!! ! !!!" ! !!!! !"#!!!!

12 plus two initial conditions!!! and!!! 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 filter13 (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 utilization of the most recent market information and the result will yield the appropriate parameters for a risk-neutral forecast of future prices.14 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 only observe 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

11 The risk premium is the amount which 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, 1995). 12 As we found it impossible to split the total estimate of!!!

! into separate estimates of!!! and!!! , our calibration procedure directly estimates!!!

! . 13 Kalman filter (Kalman, 1960) is widely used for estimating unobserved state variables and parameters (Harvey, 1989; West and Harrison, 1996). The Kalman filter produces estimates of a parameter based on measurements that contains noise or other inaccuracies. If historical spot oil prices (values for!!!) are considered as the measurement, then since!!" !! ! !! ! !!, Kalman filter methods can provide estimates of!!!’s which are normally unobservable in the market. These estimates of!!! can, in turn, be used to estimate the parameters of the equation!!!! ! !!!" ! !!!"! . 14 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 are providing assessments of the price model parameters. As the methods are based on different parameters (historical futures and options versus current futures and options), they are not directly comparable. The two approaches could conceivably be compared by looking at a specific decision situation and investigate the impact of the resulting assessments on the decision policy and its value. This is beyond the scope of this paper.

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6 SPE 160000

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 to find the value for volatility (!) that makes the Black-Scholes price match the observed option price. This approach was initially applied to options on stocks where it works as 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 lognormally distributed. The fact that market-traded futures prices are lognormally distributed allows us to write a closed form expression for valuing European put and call options on these futures contracts. Assume!!!!! is the price of a futures contract at time!!, with maturity at time!!. Following Schwartz and Smith (2000), if ! ! !"!!!!!! and the volatility of!!!"!!!!!! is !!!!!!!!, 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

! ! !!!" !!!!! ! ! !" ! ! !! !!!

(6)

where

! !!"!!! !!!! !!!

! !!! !!!

and!!!!! indicates cumulative probabilities for the standard normal distribution (! ! ! !!! ! !!). The value of a European put option with the same parameters is

! ! !!!" !!!!!! !! ! !" !! !!! ! ! !!

(7)

Step 1 - estimation of!!!!: We recorded the prices for options on futures contracts!!; prices for underlying futures contract!!!!!; time to maturity!!!; strike prices!!; and options expiry!!; as were reported in New York Mercantile Exchange on May 15 2011. The expiries of the options contracts we used are equal to the maturity of the futures contract (i.e.!! ! !). Based on these values, we solve the inverse problem to find the volatility !!!!!!!! associated with each options contract. The annualized volatility (used in the simulation of prices and real options valuation) would be!!!!!!!! !. The !!!!!!!! can be written in terms of the parameters of the two-factor model

!!! !!! ! !"# !" !!!! ! !!!! !!! ! ! !!!!"!!!

!!! !!!! ! !!!! !!! !! ! !!!"!

!!"!!!!!

(8)

If we again assume the options expire at the maturity of the futures contracts, then!! ! !, then!!!!! !!! ! !!! !!! ! !. Furthermore, it can be shown that as the maturity of the futures contracts increase (! ! !), the implied annualized volatility of futures contract will mostly reflect the uncertainty about the long-term factor15. In other words, for large T!! eq. (8) simplifies to

!!! !!!!! !!

! (9)

Thus, when maturities approach infinity, the implied annual volatility approaches the volatility of long-term factor!!! . We can insert the implied volatility of options that expire in 6-8 years; i.e.,!!! !! ! ,!!! !! ! , or!!!! !! ! , in eq. (9) and calculate!!! . Step 2 – estimation of!!!! and!!: The diagram in Figure 6 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 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!!!!! ! !!!!. Having estimated !! in the previous step, we can estimate!!!!! , the risk-neutral drift rate for the long-term factor. This curve also reveals a rough estimation for the mean-reversion coefficient!!!. The “half-life” of the deviation is the length of time that the short-term deviations are expected to halve and is equal to!!!" ! !!. The curve shows that the short-term deviations will decrease to half its value in approximately 10 months; this implies a mean-reversion coefficient of!! !" !

!!"!!"!! !!!".

15 When!! ! !,!! !"#

!!!!!!!!!"!

!! ! and!! !"#

!!!!!!!!!!"!

!! !.

SPE 160000 7

!Step 3 – estimation of!!! and!!!": When!! ! ! and we work with near-maturity futures contracts (! ! !), it can be shown that for small!!, eq. (8) results in the annualized volatility as follows16

!!! !!!!! !!!!"!!! ! !!

!! ! !!!!"!!"!!!!

(10)

The above equation shows that the annualized volatility (!! !!! !) for near maturity futures contracts (! ! !) is the sum of the volatilities of short-term and long-term factors. As we already calculated!!! in step-1, we can now insert in eq. (9) the implied volatility of options that expire; e. g., in one to three months, resulting in a system of two equations with two unknowns from which we can calculate !! and!!!!". Any two of!!!! !!!"! !!!" ,!!! !!!! !!! , or!!! !!!! !!! , inserted in eq. (3) will result in such system of linear equations. Figure 5 shows the implied volatility of observed options on futures prices taken from NYMEX on May 15th 2011. 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 (! ! !) options prices.17 Note that in all our analysis, always!! ! !!because we don’t have access to other combinations in the market.

Figure 5 - The implied volatility of observed options on futures in the commodity market and the fitted volatility curve Step 4 – estimation of!!!,!!!, and !!: We can estimate!!! (the long-term factor of the spot price at time!!),!!! (the deviation from the equilibrium at time!!), and!!! (the risk-premium for the short-term factor) based on the estimations performed in the previous steps and the relationships between parameters and initial state variables. In this step, the spot and futures prices

16 We used the fact that, when!! ! !, !!"#

!!!!!!!!!!"!

!! !! and!!!!"#

!!!!!!!!!"!

!! !.

17 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.

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Half-life = !"!!!!!

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!

Figure 4 - the futures curve and its relationship with the mean-reversion coefficient

8 SPE 160000

provide indirect information about these unknowns. Figure 6 depicts the relationships between the model parameters18. The log of the current spot oil price is the summation of!!! and!!!, therefore, since!!!! is observed in the market (WTI spot oil price was $99 per bbl on May 15th 2011) we can write!!! in terms of!!!

!! ! !" !! ! !!

(11)

Eq. (12) shows the relationship between!!!!!! (price of the futures contract at time!! with maturity at!!!) and other model parameters. We have estimated!!!! ,!!! ,!!,!!!, and!!!" previously. Furthermore, based on eq. (4) we can replace!!!! with!!!" !! ! !! in eq. (5). This leaves us with two unknowns !!! and!!!!.

!" !!!! ! !!!!!! ! !! ! !!!! ! ! ! !!!"!!!!! !! ! !!!!"!

!!!

!!!!!!! ! !!! ! !!!"!

!!"!!!!!

(12)

As discussed in footnote 19, the data points corresponding to the upper curve of Figure 6 are not observable in the market. The risk premiums!!! and!!! 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 only need !!!! (which is accurately estimated using the lower curve and eliminates the need for assessing!!!). The errors in the estimate of!!! shifts all the estimates of!!! up or down by a constant (!! !) with !! adjusting accordingly so as to preserve the sum !! ! !! corresponding to the log of expected spot price (Schwartz and Smith, 2000). Assume we replace!!! by!!!! ! ! for any!!, and in compensation replace!!! by !! ! ! ! and !! by!!! ! ! !. These changes will not affect the risk-neutral stochastic processes. We use this property and arbitrarily set !! ! ! in our estimations (which means!!! !!!). Then, if we use eq. (4) and (5) to calculate!!! and!!!!, 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. (12), and by setting!!! ! !, we can estimate!!! and!!! using eq. (11) and (12). The estimated parameters used in our economic analysis are shown in Table 1. Table 1 – Annual parameter estimates for the two-factor oil price model with assumed !! ! !!

Parameter – Description Estimated Value !!! Short-term increment of the log of the spot price 0.21 !! Long-term increment of the log of the spot price 4.38 !! The volatility of the long-term factor 10% !! The volatility of the short-term factor 29% !!! The risk-neutral drift rate for the long-term factor -0.5% !! The risk premium for short-term factor 0

18 Note that in Figure 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.

Time

Log (Prices)

Short-term Risk Premium (!"/#)

Deviation ("0)

Equilibrium Price ($0) !! !!!!

!!!!!"!!!!

!

Expected Spot Prices Slope = %$ + &'$2

Futures Curve Slope = %$ – !$ + &'$2

Spot Price

Figure 6 - Relationships among parameters of the two-factor price process

SPE 160000 9

!! Mean-reversion coefficient 0.83 !!" The correlation coefficient between the random increments 0.3

The parameter estimation procedure, the order of the estimates, and the relationships used is summarized in Table 2.

Table 2 – Summary of parameter estimation procedure Step 1 - estimation of!!!! Calculate implied volatilities for a long-maturity futures contract

using Eq. (6) or (7); then use Eq. (9) and calculate!!!! .

Step 2 – estimation of!!!! and!! Build the log of futures curve based on the observed prices of futures contracts; calculate the slope of the log of futures curve; estimate!!!!

! by subtracting!!!!!! from the slope. Estimate half-life from the futures curve; use the relationship half-life ! !" ! !! to estimate!!.

Step 3 – estimation of!!! and!!!" Calculate implied volatilities for at least two near-maturity futures contracts using eq. (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!!! and!!!".

Step 4 – estimation of!!!,!!!, and !! Arbitrarily set!!!! ! !; use eq. (11) and (12) to build a system of equations; solve the system of equations and calculate !! and!!!!

4 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 Schwartz and Smith’s two-parameter stochastic price process. We also illustrated an implied approach for calibration of 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, as 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 many 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. References Baker, M. P., E. S. Mayfield, and J. E. Parsons, 1998, Alternative Models of Uncertain Commodity Prices for Use with Modern Asset Pricing Methods, the Energy Journal, 19(1): 115-148

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