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.
Thinking in terms of real options
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. 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 real option strike price the ability to abandon or sell the property.
A property make money right now from scratch 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 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 real option strike price 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.
Why real options are important
The following equations use the algebraic symbols of Black and Scholes rather than the more modern symbols. 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.
All of these aspects make the calculation of real option value considerably more complicated than calculating the value of a financial option.
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They end their paper with the warning " 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 real option strike price, 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 real option strike price 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.
Real options analysis
One consequence of the assumptions in a Wiener process, as stated in Dixit and Pindyck page 65is 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.
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 option dissertation 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.
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The equations can be solved by a number of techniques including finite difference techniques, as those discussed by Trigeorgis pages through or by a "binomial lattice" technique, as discussed by Copeland and Antikarov in Chap.
Winston   presents a number of examples of valuing options using the Black-Scholes method and simulation. Trigeorgis pages through presents a log-transformed binomial lattice approach and gives references to other approaches from polynomial approximation to numerical integration.
Paddock, Siegel, and Smith  applied real option valuation techniques to 21 tracts in the federal lease sale number 62, held in Paddock, Siegel, and Smith used the work of Gruy, Garb, and Wood  to estimate developed reserve prices as one-third of crude oil prices.
- A real option is an economically valuable right to make or else abandon some choice that is available to the managers of a company, often concerning business projects or investment opportunities.
- Types of real options[ edit ] Simple Examples Investment This simple example shows the relevance of the real option to delay investment and wait for further information, and is adapted from "Investment Example".
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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. Data on actual transactions, as reported by Cornerstone Ventures, L. Table 1, from the Cornerstone "Annual Reserves Report," shows the annual averages for WTI and the median price for oil-dominated transactions from to Paddock, Siegel, and Smith calculate a standard deviation for the "real CPI deflated refiner cost of imported crude oil" of 0.
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 throughthe standard deviation of the change in median acquisition price is 0. This is in remarkable agreement with that calculated by Paddock, Siegel, and Smith covering the period of through Pickles and Smith calculated a standard deviation of 0.
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 through They discuss a "unit root test" to determine "whether a price series is mean reverting or is a random walk.
Dixit and Pindyck report that Wey used a year series for the real price of crude oil and found that oil prices are mean reverting and not a random walk. 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 real option strike price decisions, if the value increase justifies the cost of the information. Secondly, adhere rigorously to the converse, i. 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. The frame shift leads to a richer assessment of the opportunity. The methodology leads to procedures and presentations that can inhibit insightful discussions for key assumptions and choices. Hooper and Rutherford  also discuss the benefits of real options in framing the problem and the questions. 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.
Oil, real option strike price example, has sonic testing [sic] 2D and 3Ddrilling, and development via construction of refineries [sic], pipelines, and storage facilities. It remains to be seen how widely the process will be applied.
In his book, Trigeorgis page 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.