Real options analysis

In the last quarter-century, financial options such as "calls" and "puts" on publicly traded stocks have become an integral part of managing stock portfolios. The seminal work on financial options was done by Black and Scholes, published in 1973, and Merton, also published in 1973. Merton and Scholes shared the 1997 Nobel Prize in economics for their work. Black, Scholes, and Merton all worked on attempting to determine the value of an option. In recent years, the concepts of valuing options have been expanded from financial options to what are called "real" options in project evaluation.

Financial options

Financial options include "calls" in which the owner of the option has the right, but not the obligation (thus, an option), to purchase a stock at a specified strike price. If the option can only be exercised at the end of a specified period of time, the option is referred to as a European option. If the owner of the option can exercise the option at any time up to the expiration date, the option is referred to as an American option. A financial put option is the right, but not the obligation, to sell a stock at a specified strike price. Again, there can be European put options that are exercisable only on a specific date or American put options that can be exercised anytime prior to the expiration date.

Real options valuation/real options analysis

The proponents of real options valuation (ROV) or real options analysis (ROA) argue, for example, that "option pricing methods are superior to traditional DCF (discounted cash flow) approaches because they explicitly capture the value of flexibility." Copeland and Antikarov even go so far as to assert, "...the net present value technique systematically undervalues everything because it fails to capture the value of flexibility." There is a certain amount of irony in comparing the assertions of ROV proponents in which traditional methods undervalue everything with the assertions of portfolio analysis proponents in which "conventional treatments of uncertainty contribute to overestimation of returns..." (See Brashear, Becker, and Faulder, page 21.)

There are many types of real options that can be modeled as calls and/or puts or combinations thereof. In the oil/gas production business, the option to develop a field is similar to a call option. The producer has the option to invest the development costs and receive the value of the reserves. An example of a put option is the case in which the producer has the ability to abandon or sell the property. A property sale differs from a stock put in that the price of the sale might be unknown, while the exercise price of a stock put is usually known with certainty. There are a number of other types of real options. Trigeorgis lists several types of real options including:

• The option to defer investment
• The option to default during staged construction
• The option to expand
• The option to contract
• The option to shut down and restart operations
• The option to abandon for salvage value
• The option to switch use
• The corporate growth option

Copeland, Koller, and Murrin list similar real options along with compound options, which are options on options, and "rainbow" options in which there are multiple sources of uncertainty. As they state, the exploration and development of natural resources is an example of a compound rainbow option. All of these various types of options can be valued if several parameters are known. The simplest place to begin is with a call option.

The most famous equation in option valuation is the Black-Scholes equation for a European call option. The following equations use the algebraic symbols of Black and Scholes rather than the more modern symbols.

....................(1)

where N(d) is the cumulative normal density function and

....................(2)

and

....................(3)

In Eq.1, w(x,t) is the value (at any time, t) of the call option on a stock with a current price of x.

• The strike or exercise price is c
• The risk-free interest rate is r
• The maturity date is t*
• v² is the variance rate of the return on the stock

The variance is one of the most important and interesting parts of the equation. Black and Scholes assume, among other things, that "the distribution of possible stock prices at the end of any finite interval is log-normal" and that the stock price on any day is independent of the price on the previous day. This assumption of a "random-walk" in stock price is premised on the existence of an efficient market in which the stock is fairly valued on any given day, and all the information available concerning the stock has been taken into account by the market.

While the Black-Scholes formula revolutionized the financial markets, it has had little direct application in the oil/gas business because the assumptions used in its development are not particularly appropriate for oil/gas properties. As Davis* explains, "real options are different from financial options because the exercise price (in both calls and puts) is not known with certainty; exercise is not instantaneous; and the stochastic process for the underlying asset is not the same as it is for financial options. All of these aspects make the calculation of real option value considerably more complicated than calculating the value of a financial option."

Lohrenz and Dickens present a real-world oil/gas example using the Black Scholes formula in their comparison of option theory and discounted cash flow methods for an actual field in the offshore Gulf of Mexico. They discuss many options that are available during the "lifetimes of searchable, developable, and producible oil/gas assets." In their analysis, they showed that the value of the development option (like a call option on a stock) increased dramatically (by a factor of greater than 3) as the variance in the oil/gas asset value increased from zero (perfect knowledge) to 1.0/year. They end their paper with the warning "...we should always temper results from uncertainty analyses [both option theory and decision trees] and their use with the understanding that the real world and its real uncertainties have not been captured—only modeled by necessarily flawed and incomplete practice and practitioners." Another good example of the Black-Scholes model is presented by Copeland and Antikarov (pages 106 through 110).

Mathematics and applications of real options

The mathematics and application of real options can quickly become very complex. The value of the asset underlying the option is assumed to vary in time in a stochastic manner. That means at least part of the price varies in a random and unpredictable fashion. Dixit and Pindyck discuss several potential mathematical models for the price of the asset beginning on page 59 of their book. The most common of these in financial options is the Wiener or Brownian motion process. In this process, the change in value from one period to the next is assumed to follow a normal or log-normal distribution. (The standard assumption for stock prices is that the change in price over time is log-normally distributed.) This process is one of the underlying assumptions in the Black-Scholes model. As Dixit and Pindyck point out, the process has some interesting properties—one being that all future values depend only on the current value and not on any historical value. In other words, there is no memory of past prices. One consequence of the assumptions in a Wiener process, as stated in Dixit and Pindyck (page 65), is that "the variance of the change in a Wiener process grows linearly with the time horizon" and that "over the long run its variance will go to infinity." This is problematical when applying this process to the value of oil/gas properties. There is certainly an upper limit to the value of an oil/gas property no matter what the time frame. Those who lived through the "boom" in the late 1970s will remember the commonly accepted forecasts showing average annual oil prices of $100/barrel or more before the end of the 21^st century.

Other potential mathematical options

Dixit and Pindyck discuss a number of other potential mathematical models including:

• "Brownian motion with drift" (a process with an increasing or decreasing trend and randomness superimposed)
• "mean-reverting processes,"
• jump processes (sometimes called Poisson processes)

Applied processes

When these processes are applied to value-an-option on an underlying asset of a second-order partial differential equation results. These equations, just like the diffusivity equation common in fluid flow in porous media, only have analytical solutions for certain simple boundary conditions (such as those used by Black and Scholes). The equations can be solved by a number of techniques including finite difference techniques, as those discussed by Trigeorgis (pages 305 through 320) or by a "binomial lattice" technique, as discussed by Copeland and Antikarov in Chap. 7 of their book where they present a spreadsheet model for a binomial lattice. Winston presents a number of examples of valuing options using the Black-Scholes method and simulation. Winston’s book contains a CD with spreadsheets for all examples. Trigeorgis (pages 320 through 329) presents a log-transformed binomial lattice approach and gives references to other approaches from polynomial approximation to numerical integration. Pickles and Smith also discuss the binomial lattice method and present a numerical example for producing oil/gas properties.

Paddock, Siegel, and Smith applied real option valuation techniques to 21 tracts in the federal lease sale number 62, held in 1980. This study is also used as an example for oil/gas in the text by Dixit and Pindyck (pages 396 through 403) and is cited by Trigeorgis as "the first empirical evidence that option values are better than DCF-based bids in valuing offshore oil leases."

Valuing a developed reserve, even at a particular time, is quite difficult. Paddock, Siegel, and Smith used the work of Gruy, Garb, and Wood to estimate developed reserve prices as one-third of crude oil prices. In practice, the value of developed reserves is highly dependent on the perceptions of future prices, as well as the level of operating costs and the fiscal terms. Johnston (pages 13 and 14) estimates that "proved, developed, producing reserves are worth from one-half to two-thirds of the wellhead price times the contractor’s take." The United States has a contractor’s take of about 50%, so producing reserves (in this case, working interest reserves) would be worth about one-fourth to one-third of the wellhead price using Johnston’s rule of thumb. If contractor’s take is lower, such as the approximately 10% contractor’s take in Venezuela, then, obviously, the working interest reserves would be worth much less. Data on actual transactions, as reported by Cornerstone Ventures, L.P. for the period of 1991 through 1998, indicate that the median price of U.S. proved developed producing reserves, in the ground and net of royalty, during that time period ranged from $4.08/bbl to $5.26/bbl, while the yearly average spot price of oil (WTI) ranged from $14.37/bbl to $22.20/bbl. Table 1, from the Cornerstone "Annual Reserves Report," shows the annual averages for WTI and the median price for oil-dominated transactions from 1991 to 1998. The ratio of the median price to the WTI price ranges from 19 to 28% with an average of about 25%. Paddock, Siegel, and Smith calculate a standard deviation for the "real (CPI deflated) refiner cost of imported crude oil" of 0.142/year. They then assumed that this standard deviation would apply to the change in value of the underlying asset (reserves in the ground). An analysis of the Cornerstone figures indicates that, for the period of 1991 through 1998, the standard deviation of the change in median acquisition price is 0.15/year on either a nominal basis or on a CPI adjusted basis. This is in remarkable agreement with that calculated by Paddock, Siegel, and Smith covering the period of 1974 through 1980. Pickles and Smith calculated a standard deviation of 0.22/year for the period of 1985 through 1989 based on quarterly median reserve prices as reported by Strevig and Associates, the predecessor to Cornerstone Ventures. These values for standard deviation would appear to be in good agreement, but one of the underlying assumptions is that property prices follow a Wiener or Brownian motion process.

• 

  Table 1

Underlying process for oil prices

One of the more disconcerting aspects of this assumption is the discussion of the underlying process for oil prices as discussed by Dixit and Pindyck (pages 403 through 405). They discuss a "unit root test" to determine "whether a price series is mean reverting or is a random walk." In their words, "this is a weak test that for short time series (for example, 30 years or less) will often fail to reject the hypothesis of a random walk, even if the series is in fact mean reverting." Considering the different processes underlying the price of oil/gas such as the Texas Railroad Commission in the 1950s and 1960s, the rise of OPEC’s power in the 1970s, the Federal Power Commission’s regulation of natural gas prices—all overlying the fundamental supply and demand relationship, it is difficult to imagine a meaningful process lasting much longer than 30 years. Dixit and Pindyck report that Wey used a 100-year series for the real price of crude oil and found that oil prices are mean reverting and not a random walk. They report that Wey** shows "ignoring mean reversion can lead one to undervalue the reserve by 40 percent or more" when the development cost in $/bbl is one-half the mean developed reserve price, while it will have little effect when the development cost is close to the developed reserve price.

Option pricing principles applied to purchase of information

Chorn and Carr discuss option pricing principles and then apply those principles to the purchase of information. Their advice is to "purchase information that will impact the upcoming decisions, if the value increase justifies the cost of the information. Secondly, adhere rigorously to the converse, i.e., invest now or abandon the project if there is no information to be gained (or it’s [sic] expense is too great) that will significantly change the project’s outcome or impact the investment decision process." This is sound advice with or without real option analysis, but they show how real options can be used to value that information—a value which is often difficult to quantify.

Benefits and difficulties with real options

Davidson presented an excellent paper on benefits and difficulties with real options. He states that "the primary contribution of ROA is to produce a frame shift. Instead of thinking about a project from a do-it or don’t-do-it frame, ROA promotes thinking from a what-are-all-the-possibilities frame. The frame shift leads to a richer assessment of the opportunity." He criticizes the approach by stating that "the ROA valuation methodology is not only inaccurate for E&P projects, it is needlessly complicated. The methodology leads to procedures and presentations that can inhibit insightful discussions for key assumptions and choices." He then presents a method of getting the benefits of both real options analysis and present value analysis. Hooper and Rutherford also discuss the benefits of real options in framing the problem and the questions.

Real options analysis in the E&P business

Real options analysis is slowly working its way into the E&P business. The mathematics are daunting; the terminology is foreign; the underlying assumptions are shaky; and communicating the results in an easily understood manner is difficult, but the method does show promise. The breach between the theoreticians and the practitioners in the E&P business needs to be bridged. This breach is well illustrated by the following quote from Copeland and Antikarov’s practitioner’s guide (page 164) in which they discuss compound options. "Exploration and development for natural resources (oil, natural gas, gold, copper, and coal) have multiple phases. Oil, for example, has sonic testing [sic] (2D and 3D), drilling, and development via construction of refineries [sic], pipelines, and storage facilities." On the other hand, the frame shift mentioned by Davidson and Hooper and Rutherford is a real benefit. It remains to be seen how widely the process will be applied. In his book, Trigeorgis (page 375) lists ten points of future research. The first two (listed next) are very relevant.

• Analyzing more actual case applications and tackling real-life implementation issues and problems in more practical detail.
• Developing generic options-based user-friendly software packages with simulation capabilities that can handle multiple real options as a practical aid to corporate planners.

If these points are done, along with educating the managers, real options could become a useful tool.

* Personal communication with Graham Davis, Colorado School of Mines, Golden, Colorado (2002).** Wey, L.: "Effects of Mean-Reversion on the Valuation of Offshore Oil Reserves and Optimal Investment Rules," unpublished undergraduate thesis, MIT, Cambridge, Massachusetts (May 1993).

Nomenclature

c	=	strike or exercise price
d1, d2	=	intermediate variables
N(d)	=	cumulative normal density function
r	=	risk free interest rate
t*	=	maturity date
t	=	time
v2	=	variance rate of the return on the stock
w(x,t)	=	value of a European call option at any time, t
x	=	current price of the stock

Noteworthy papers in OnePetro

Willigers, B. J. A., & Bratvold, R. B. (2008, January 1). Valuing Oil and Gas Options by Least Squares Monte Carlo Simulation. Society of Petroleum Engineers. http://dx.doi.org/10.2118/116026-MS

External links

Use this section to provide links to relevant material on websites other than PetroWiki and OnePetro

See also

Risk and decision analysis

Portfolio analysis

Key economic parameters for decision making

PEH:Petroleum_Economics

PEH:Risk_and_Decision_Analysis

Page champions

Russell S J Krasey - P.Eng