James A. MacKayTexaco E&P Technology Division

Houston

Ian LercheUniversity of South Carolina

Columbia, S.C.

In the stock market the option to buy a stock at a later time at some fixed price requires money.

For instance, if a stock is trading at $50/share and one anticipates the stock price will rise to $80/share, then the cost of the option to buy shares at some future time for $50 each should be no more than $30/share ($80 minus $50), because if the option is exercised the total costs per share are $50 plus the option cost of $30.

If the stock price actually drops below $50/share then the option contract to buy it at $50 is worth nothing, and one has lost $30/share.

The stock market question is: How much should one pay for the option?

Guessing correctly leads to spectacular financial gain, and guessing incorrectly leads to financial ruin, as in the recent bankruptcies of Orange County, Calif., and Barings Bank, London.

In hydrocarbon exploration a somewhat similar option problem arises as follows. Suppose an exploration opportunity is available. Corporate assessment of the opportunity is made with the result that the probability of success (i.e., hydrocarbons being found) is estimated at ps (with a corresponding probability of failure of pf = 1 2 ps), total gains to the corporation, if the project is successful, are estimated at $G, while costs to finance the project are $C.

Conventionally, the decision to become involved in the project is based on the expected value, EV, which, for the decision tree representation (Fig. 1A)(13030 bytes), is EV = psG 2 C. A positive EV will, on average, make the project worthwhile, i.e. G $ C/ps is necessary for EV $ 0. Note that the success probability branch of the decision tree has a high channel value, HCV, of HCV = ps (G 2 C).

The reason that a corporation might want to take an option on an opportunity is to wait until more information becomes available before committing to the decision to drill. For instance, in a situation where an opportunity is one of the first prospects in a new play trend, there is often very little known (as opposed to surmised) about the potential gain, and the chances of success are also often poorly determined. Thus, until the estimates of costs, gains, and success chances are firmed up in the future by drilling information from other operators, the corporation would prefer to hold off on the decision to drill a particular prospect.

On the other hand, the corporation must make a decision now, on the basis currently available information, as to the maximum amount it will pay for the exclusive option to drill later. Such option costs then add to the total project cost if the corporation eventually decides to drill the prospect, thereby reducing estimated gains, unless future information turns out to provide a massive shift in the estimates of chances of success or gains.

However, if the current estimated chance of success is already very close to 100%, then one would not want to pay very much for an option because it is highly unlikely that future information will drive the success probability down very far; one would in that instance likely prefer to drill the prospect. In addition, if the estimated cost of development as well as reserve uncertainty are both very much less than the uncertainty in gains, then one would also likely prefer to drill the prospect.

Equally, if the expected value, EV, of the project is small, then paying a large option price will drive down the expected value (including both option costs and project costs) and, if the option paid exceeds EV, the new expectation is for a negative EV, so that the project is unlikely to ever prove profitable.

Thus, the maximum option worth, based on available information, is limited at both the extreme of a highly successful chance of success and the extreme of a low expected value. The problem is to evaluate how the maximum option worth depends on currently available information, so that a future decision to drill or not drill provides the best return to the corporation when one does not yet have available the futureward information.

Recently Dixit and Pindyck1 have persuasively argued qualitatively that an options approach should be taken to capital investment in any business field because, Opportunities are optionsrights but not obligations to take some action in the future and, as soon as you begin thinking of investment opportunities as options, the premise (that investment decisions can be reversed if conditions change or, if they cannot be reversed, that they are now-or-never propositions) changes. Irreversibility, uncertainty, and the choice of timing alter the investment decision in critical ways.

From an oil industry perspective Dixit and Pindyck eloquently provide the rationale for considering an option position rather than dealing only with net present value considerations.

The ability to calculate the worth of an option to a hydrocarbon exploration opportunity is what this article is about.

### Option amounts

High gain scenario. The optioning position that a corporation can take with respect to the opportunity can be set up as follows:

Let the corporation be prepared to pay up to a maximum option amount, Ow, for the privilege of investing in the opportunity at some future time with a buy-in working interest of 100% and total project costs of C. Then the total corporate costs, if the option is exercised, are C 1 Ow, while the gains remain fixed at G. Thus the decision-tree diagram is then modified (Fig. 1B)(13030 bytes) and the expected value to the corporation is reduced to EV1 = (psG 2 C) 2 Ow.

The expected high gains to the corporation are given by the success probability branch of the decision-tree of Fig. 1B (13030 bytes), namely HCV

ps [(G 2 C) 2 Ow]. The maximum option worth, Ow, is then just the difference between HCV and EV1, provided both HCV $ 0 and EV1 $ 0. Then

Ow = ps [(G 2 C) 2 Ow] 2 [(ps G 2 C) 2 Ow] (1)

which gives the maximum option worth explicitly as

Ow = C (1 2 ps)/ps (2)

provided

G $ C/ps (3)

which is necessary to keep EV1 $ 0. Note that the gains cancal out of expression (2) for Ow, directly reflecting the fact that it is the increased costs that lower HCV and EV1. When the maximum option worth is given by equation (2) it follows by direct substitution of equation (2) into the formula EV1 = (ps G 2 C) 2 Ow that the expected value of the project is

EV1 = ps (G 2 C/ps ) (4a)

and the success probability branch worth is then given by

HCV = ps (G 2 C/ps) (4b)

If the option is exercised at the maximum option worth, the total project costs (TPC) to the corporation are then given by

TPC = C 1 Ow = C/ps (5)

Partial option amounts. The estimates of option worth so far have been on the assumption that the corporation is prepared to pay the maximum option worth, Ow, and that G is sufficiently large to still support that level of cost. However, a corporation may balk at the total option price and be prepared to pay only a fraction f of the maximum Ow for the same working interest of 100% in an opportunity of project costs C. In such a situation the expected value is E2 = (ps G 2 C) 2 fC pf/ps, while the success probability branch yields HCV2 = ps(G 2

C) 2 fC(1 2 ps). Then E2 $ 0 and HCV2 $ 0 as long as f # 1 2 pf1 [1 2 ps G/C] in ps G # C, and f # 1 in ps G $ C which requires only that ps G . C to yield E2 $ 0. The extra requirement that f be a fraction less than unity implies

f # min{1,1 2 pf1 (1 2 ps G/C)}

For instance, consider the situation in which ps

f # min {1,1 2 pf1 (1 2

4ps)}

Note that 1 2 pf 1 (1 2 4ps) takes on the value 1 when ps = 12 and, for values of ps less than 12, f # 1 2 pf1 (1 2 4ps). As ps decreases, the fraction of the maximum option worth that the corporation can take drops to zero at ps = 14, which is where psG = C, and zero expected value EV then occurs.

Thus the corporate maximal option position is then restricted to fOw, which limitation is an option commitment of less than or equal to [1 2 pf1 (1 2 ps G/C)]Ow in ps G # C. Effectively the corporation can option up to an amount psG 2 C (i.e., the expected value) in G

### Numerical illustrations

Current value option worths. Consider an exploration opportunity in which the potential gains, G, are estimated at $100 MM, project costs, C, are estimated at $10 MM, the success probability is 50% (ps = 0.5), and in which the corporation is prepared to take a 100% working interest. The question is what is the maximum option worth, Ow? First note that Gps/C = 2.5, which exceeds unity, so that an optioning position up to the maximum can indeed be taken.

Then the maximum option worth is Ow 5 $10 3 (1 2 0.5)/0.5 MM = $10 MM; the total project costs to the corporation if the option is exercised are TCP = $(10 1 5) MM = $15 MM, where $10 MM represents 100% of the option value and $5 MM represents the probability of success times original project costs; while the expected value is EV1 = $30 MM, and HCV = $40 MM.

If no option position had been taken (i.e., if Ow 0)

then EV = $40 MM, so that the option exercise has reduced EV from $40 MM to $30 MM; but HCV (with Ow = 0) would then have been $45 MM. Thus, if one did not include the maximum option worth in the assessment of HCV or EV, then one would have had $(45-40) MM $5 MM as an estimate of the maximum option worth. In fact, however, the option is maximally worth $10 MM, because HCV and EV are reduced by inclusion of option costs, but EV is reduced faster than HCV, thereby increasing the option worth.

If the corporate assessment of the opportunity reflects the likely success probability, gains, and costs accurately, then the corporation should be prepared to pay up to a maximum of $10 million for the option to buy-in to the opportunity at 100% working interest at a future time. The option worth of $10 million is only 25% of the estimated HCV of $40 million and only 3313% of EV1, so that the option costs are a small fraction of potential gains.

Because option worth is sensitive to costs, success probability and gains (through the requirement G $ Cps to option at the maximum), and because ps, G, and C are uncertain at the assessment stage of an exploration opportunity, it makes corporate sense to evaluate the sensitivity of maximum option worth to variations in gains, costs, and the chance of success.

In Fig. 2 (14923 bytes) is plotted fOw (in $MM) as the gains, G (also in $MM), vary for fixed values of ps = 0.5 and costs, C = $10 MM. Note that in G # $40 MM the maximum option worth cannot be taken because then one would have EV1 # 0. Accordingly the option worth is restricted to less than or equal to ps G 2 C in G

In Fig. 3 (15010 bytes) is plotted the option worth fOw (in $MM) as the costs of the project vary for a fixed value of ps = 0.5, and for a fixed gain of G = $40 MM. Note that the requirement G $ C/ps is then only satisfied for C # $10 MM, so that the maximum option worth increases linearly with increasing costs at low costs (reflecting directly the relative increase of HCV compared to EV1) but drops to ps G 2 C at C $ $10 MM.

In Fig. 4 (14778 bytes) is plotted the option worth fOw (in $MM) as the success probability systematically increases for fixed values of gains, G = $40 MM, and costs, C = $10 MM. In this case note that the worth of the option is ps G 2 C for ps # (C/G)1/2 = 0.5 and, at higher values of ps, the maximum option worth declines with increasing ps from a value of $10 MM at ps = 0.5 because, as ps 1, there is less and less possibility of a failure, so that the difference between HCV and EV1 systematically lessens, making the maximum option worth that much less.

Current option value and future decisions. A prospect generator has come to a corporation with a prospect to drill. He wants his usual override and would like the well spudded in three months (before other key acreage is acquired and drilled by others). Because this prospect is one of the first in a new play trend, very little is known about the potential gain, but it is estimated to range from a 10% chance of more than $200,000 to a 90% chance of less than $2 million after post drilling costs and the override. The mean estimate is $1 million and the mode is $300,000. The cost to drill is $100,000, and the chance of success is estimated at 35%. Although the expected value is $250,000, based on the mean value (if successful) of $1 million, it is most likely that for this one prospect (as with any individual venture) that the mode of $300,000 will be the result if successful.

Rather than take the prospect as proposed and commit to early drilling, the corporation offers $100,000 for the option to drill later during a five year lease term. The corporation could have offered up to $185,000 based on the calculated maximum option value of (Cx (1 2 ps))/ps. The option is accepted.

The corporation then waits three years, and in the meantime, several wells are drilled on trend that indicate the range of the gain is now $500,000 to $2.5 million, with a mean of $1.5 million and a mode of $1 million. The drilling costs are higher at $150,000 and the chance of success is higher at 50%.

The corporation could drill the well based on the new prospect EV of $600,000, or an EV (including the option cost) of $500,000. The well could come in at the modal value of $1 million, and there is a 50% chance of making $750,000. However, a successful operator in the area (with even more information about chance, cost, and size) offers the corporation $500,000 for the option. The question is: what is the best decision to make based on both the initial information and the three year later information in order to maximize gains to the corporation in relation to total costs?

Four possibilities need to be considered. With the initial information supplied by the prospect generator the corporation could choose:

(i) to drill immediately and not option;

(ii) to option, but then to drill immediately;

(iii) to option, but to wait three years and then drill;

(iv) to sell the option at the end of the three years.

Shown in the table are the ratios of return to costs (12464 bytes) for the four different scenarios. It is clear from the table that there is little point in optioning and then drilling immediately; effectively one has just doubled the cost of the project by doing so. Thus possibility (ii) is to be avoided.

Between possibilities (i) and (iii) it is apparent from the table that the information that becomes available at the three year stage is not sufficient to offset the price of $0.1 MM paid initially for the option. The ratio of earnings/total costs drops under possibility (iii) relative to possibility (i), implying that one paid too much for the option initially relative to the improvement in knowledge gained three years later.

Possibility (iv) has the highest ratio of earnings/total costs, implying that one makes the right decision in selling the option to the more successful operator. If one had chosen not to sell the option then one would have been better off drilling the prospect from the start without paying an option cost of $0.1 MM.

A different point of view is to note that successful wells on the trend over the three year period have typically had a 50% success rate and have had drilling costs of around $0.15 MM. Given that one did commit $0.1 MM to the option position three years ago, what sort of potential gains are needed now in order to make the earnings/cost ratio larger than it would have been if one had just drilled the prospect initially? Because

EV = G $0.25 MM,

while project costs plus option are at $0.25 MM, it follows that the ratio of EV/costs, if one had drilled directly, would be 2.5 (see table) so that if

1 G 2 0.252 /0.25

is to exceed 2.5, potential gains in excess of $1.75 million are required to be estimated now based on the later information.

An alternative viewpoint is to ask a slightly different question: If, at the initial stage, one anticipates that the success probability will rise to 0.5, the gains to increase to $1.5 million, and the costs to increase to $150,000 in the three years, what should the option price be in order that earnings/total cost for possibility (iii) be larger than for possibility (i)?

The new expected value, including an option cost amount Ow (in million dollars), with ps = 0.5, G = $1.5 MM, and project costs of $0.15 MM, is EV1 = $0.6 MM Ow; the total costs of the project (including the option cost) are now C1 = $0.15 MM + Ow. Thus the ratio of earnings/cost is EV1/C1 = (0.6 Ow)/(0.15 + Ow). If EV1/C1 is to exceed the original value of the ratio of EV (= $0.25 MM) to costs ($0.1 MM) (i.e. 2.5) it then follows that Ow # $64,300. Thus, at an investment of $100,000 one overpaid for the option relative to anticipated improvements in knowledge.

Clearly, the aim of the option decision at the initial stage is not only to figure out the maximum option worth one could pay based on then available information, but also to estimate an actual option price (or range) that one should pay if future information is to lead to an improved ratio of earnings/costs.

The difficulty, as usual, is that one does not have available the futureward information at the time the option decision has to be made. Thus one has to evaluate the option amount either on the basis of then available information or on what if potential future scenarios, in order to estimate the likelihood of an option being worthwhile.

### Discussion

The point of these illustrations has been to show that it is relatively simple to figure the maximum worth of an option to an exploration opportunity, and also relatively easy to examine sensitivity of the maximum option worth to changes in parameter values of the opportunity influencing the option worth.

The rapid variation of the maximum option worth as gains, costs, and success probability all vary, and the shaping of the option worth curve with each variable, provide indications of the sensitivity, and so of the chances for guessing correctly (high financial gain) or incorrectly (financial loss of at least the option worth) the likely worth of an exploration opportunity.

Inclusion can also be given of a working interest less than 100%, as well as of both a corporate risk tolerance and a fixed budget, in an examination of maximum option worth in exploration opportunities, and also of portfolio balancing of opportunities to maximize total risk adjusted value, as well as of partial option worth.2

In addition, one can also include the ability to option in such a way that the expected value after optioning cannot be less than the expected value before optioning (in which case the maximum option amount is Cpf) rather than requiring only a positive EV as we do here.

While such considerations are of concern in assessing corporate strategy vis-a-vis a portfolio of opportunities; and while fluctuations in gains, costs, and success probabilities can also be included to determine where the greatest degrees of sensitivity lie in a given portfolio and the associated straddle of probable option worths; a detailed discussion of those points would make for a very long article indeed, and one that would defeat the purpose of presenting simply the essence of the worth of an option position for an exploration opportunity, which is the main point of this article.

### Acknowledgments

The work reported here was supported by the Industrial Associates of the Basin Modeling Group at USC, and especially by Texaco.

### References

- Dixit, A.K., and Pindyck, R.S., The options approach to capital investment, Harvard Business Review, issue of May-June 1995, pp. 105-115.
- MacKay, J.A., and Lerche, I., On the value of options in exploration economics, submitted to Energy Exploration and Ex- ploitation, 1996.

### The Authors

Jim MacKay has worked for Texaco since 1968 as a geologist, geophysicist, planning manager, exploration manager, and now career development manager. He has taught dozens of courses for Texaco and AAPG covering risk assessment and decision analysis and received a Texaco award for outstanding instructor in 1993.

He has a degree in geology from Brigham Young University.

Ian Lerche has been a professor of geology in the department of geological sciences at the University of South Carolina since 1984. He was an associate chairman of the geology department from 1985-89. From 1965-81 he held the positions of research associate, assistant professor, and then associate professor at the University of Chicago.

From 1981-84 he worked as a senior research geophysicist and then as a research associate for Gulf Research & Development Co. He received a BS in physics in 1962 and a PhD in astronomy in 1965 from the University of Manchester.

*Copyright 1995 Oil & Gas Journal. All Rights Reserved.*