Real options valuation

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value of a real option (time, discount rates, volatility, cash inflows and outflows) are each affected by the terms of business, and external environmental factors that a project exists in. Terms of business as information regarding ownership, data collection costs, and patents, are formed in relation to political, environmental, socio-cultural, technological, environmental and legal factors that affect an industry. Just as terms of business are affected by external environmental factors, these same circumstances affect the volatility of returns, as well as the discount rate (as firm or project specific risk). Furthermore, the external environmental influences that affect an industry affect projections on expected inflows and outlays.

155:(5M) are greater than the investment costs (4M) by 1M. Yet, if the firm waits for next year, it only invests if discounted cash flows do not decrease. If discounted cash flows decrease to 3M, then investment is no longer profitable. If, they grow to 6M, then the firm invests. This implies that the firm invests next year with a 66.7% probability and earns 5.45M - 3.63M if it does invest. Thus the value to invest next year is 1.21M. Given that the value to invest next year exceeds the value to invest this year, the firm should wait for further information to prevent losses. This simple example shows how 4642: 4632: 67:, and must instead rely on their perceptions of uncertainty. Unlike financial options, management also have to create or discover real options, and such creation and discovery process comprises an entrepreneurial or business task. Real options are most valuable when uncertainty is high; management has significant flexibility to change the course of the project in a favorable direction and is willing to exercise the options. 5560: 379:. Here, observing the outcomes relating to the first project, the firm can resolve some of the uncertainty relating to the venture overall. Once resolved, management has the option to proceed or not with the development of the other projects. If taken in parallel, management would have already spent the resources and the value of the option not to spend them is lost. The sequencing of projects is an important issue in 562:

volatility may limit the value of an option.) Part of the criticism (and subsequently slow adoption) of Real Options Valuation in practice and academe stems from the generally higher values for underlying assets these functions generate. However, studies have shown that these models are reliable estimators of underlying asset value, when input values are properly identified.

525:. Because management adapts to each negative outcome by decreasing its exposure and to positive scenarios by scaling up, the firm benefits from uncertainty in the underlying market, achieving a lower variability of profits than under the commitment/NPV stance. The contingent nature of future profits in real option models is captured by employing the techniques developed for 981:, the market and environment underlying the project must be one where "change is most evident", and the "source, trends and evolution" in product demand and supply, create the "flexibility, contingency, and volatility" which result in optionality. Without this, the NPV framework would be more relevant. 204:

By opening one store, the firm knows that the probability of high demand is 50%. The potential value gain to expand next year is thus 50%*(10M-8M)/1.1 = 0.91M. The value to open one store this year is 7.5M - 8M = -0.5. Thus the value of the real option to invest in one store, wait a year, and invest

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Consider a firm that has the option to invest in a new factory. It can invest this year or next year. The question is: when should the firm invest? If the firm invests this year, it has an income stream earlier. But, if it invests next year, the firm obtains further information about the state of the

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Real options are generally distinguished from conventional financial options in that they are not typically traded as securities, and do not usually involve decisions on an underlying asset that is traded as a financial security. A further distinction is that option holders here, i.e. management, can

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in 1997. Arundel involves a group of investors that is considering acquiring the sequel rights to a portfolio of yet-to-be released feature films. In particular, the investors must determine the value of the sequel rights before any of the first films are produced. Here, the investors face two main

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can (should) be applied instead, here valuing the option to "exchange" expenses for revenue. (Relatedly, where the project is exposed to two (or more) uncertainties — e.g. for natural resources, price and quantity — some analysts attempt to use an overall volatility; this, though, is more correctly

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When valuing the real option, the analyst must therefore consider the inputs to the valuation, the valuation method employed, and whether any technical limitations may apply. Conceptually, valuing a real option looks at the premium between inflows and outlays for a particular project. Inputs to the

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right on a product has a right to develop and market the product exclusively until the expiration of the patent. The firm will market and develop the product only if the present value of the expected cash flows from the product sales exceeds the cost of development. If this does not occur, the firm

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itself, is the right—but not the obligation—to undertake certain business initiatives, such as deferring, abandoning, expanding, staging, or contracting a capital investment project. For example, real options valuation could examine the opportunity to invest in the expansion of a firm's factory and

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is often not tradable – e.g. the factory owner cannot easily sell the factory upon which he has the option. Additionally, the real option itself may also not be tradeable – e.g. the factory owner cannot sell the right to extend his factory to another party, only he can make this decision

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The relevance of Real options, even as a thought framework, may be limited due to market, organizational and / or technical considerations. When the framework is employed, therefore, the analyst must first ensure that ROV is relevant to the project in question. These considerations are as follows.

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First, you must figure out the full range of possible values for the underlying asset.... This involves estimating what the asset's value would be if it existed today and forecasting to see the full set of possible future values... calculations provide you with numbers for all the possible future

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and investment projects. These help quantify the value of flexibility engineered early on in system designs and/or irreversible investment projects. The methods help rank order flexible design solutions relative to one another, and thus enable the best real option strategies to be exercised cost

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suggests the firm should not invest: the net present value is -0.5M per store. But is it the best alternative? Following real options valuation, it is not: the firm has the real option to open one store this year, wait a year to know its demand, and invest in the new store next year if demand is

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allow for flexibility as to exercise, where the relevant, and differing, rules may be encoded at each node. Note that lattices cannot readily handle high-dimensional problems; treating the project's costs as stochastic would add (at least) one dimension to the lattice, increasing the number of

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leads to higher value. (An application of Real Options Valuation in the Philippine banking industry exhibited that increased levels of income volatility may adversely affect option values on the loan portfolio, when the presence of information asymmetry is considered. In this case, increased

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that the real option and the underlying project are both traded: the so called, Marketed Asset Disclaimer (MAD) approach. Although this is a strong assumption, it is pointed out that a similar fiction in fact underpins standard NPV / DCF valuation (and using simulation as above). See:

3990:, Keith Leslie and Max Michaels McKinsey Quarterly, 1997 (3) pages 4–22. Cited by Robert Merton in his Nobel Prize Acceptance Speech in 1997. McKinsey classic - Reprinted in McKinsey Anthology 2000 - On Strategy. Cited in McKinsey Anthology 2011 - Have You Tested Your Strategy Lately. 219:

The flexibility available to management – i.e. the actual "real options" – generically, will relate to project size, project timing, and the operation of the project once established. In all cases, any (non-recoverable) upfront expenditure related to this flexibility is the

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exercise at any point (many points) in the project's life and are impacted by multiple underlying variables, the standard methods are limited either with regard to dimensionality, to early exercise, or to both. In selecting a model, therefore, analysts must make a

287:: Here the project is designed such that its operation can be dynamically turned on and off. Management may shut down part or all of the operation when conditions are unfavorable (a put option), and may restart operations when conditions improve (a call option). A 874:

are sometimes applied. Although many of the early ROV articles discussed this method, its use is relatively uncommon today—particularly amongst practitioners—due to the required mathematical sophistication; these too cannot readily be used for high-dimensional

1176:, and modify their models correspondingly. Under this approach, (a) we "replicate" the cash flows on the option by holding a risk free bond and the underlying in the correct proportions. Then, (b) since the cash flows of the option and the portfolio will 1042:(some real options, however, can be sold, e.g., ownership of a vacant lot of land is a real option to develop that land in the future). Even where a market exists – for the underlying or for the option – in most cases there is limited (or no) 169:

Staged investments are quite often in the pharmaceutical, mineral, and oil industries. In this example, it is studied a staged investment abroad in which a firm decides whether to open one or two stores in a foreign country. This is adapted from

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articles: "In financial terms, a business strategy is much more like a series of options, than a series of static cash flows". Investment opportunities are plotted in an "option space" with dimensions "volatility" & value-to-cost ("NPVq").

407:. These options are particularly valuable in industries where demand is volatile or where quantities demanded in total for a particular good are typically low, and management would wish to change to a different product quickly if required. 417: – allows management to use different inputs to produce the same output as appropriate. For example, a farmer will value the option to switch between various feed sources, preferring to use the cheapest acceptable alternative. An 989:

Real options are "particularly important for businesses with a few key characteristics", and may be less relevant otherwise. In overview, it is important to consider the following in determining that the RO framework is applicable:

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scenarios for the projection of the future pay-off distribution, and are not based on restricting assumptions similar to those that underlie the closed form (or even numeric) solutions discussed. Recent additions include the

450:, and we would therefore expect options-based modelling and analysis to be applied here. At the same time, it is nevertheless important to understand why the more standard valuation techniques may not be applicable for ROV. 552:

Given these different treatments, the real options value of a project is typically higher than the NPV – and the difference will be most marked in projects with major flexibility, contingency, and volatility. As for

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per store is 7.5M. It is also known that if the store's demand is independent of the store: if one store has high demand, the other also has high demand. The risk neutral rate is 10%. The investment cost per store is 8M.

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Given the similarity in valuation approach, the inputs required for modelling the real option correspond, generically, to those required for a financial option valuation. The specific application, though, is as follows:

371:: This option is related to the initiation option above, although entails flexibility as to the timing of more than one inter-related projects: the analysis here is as to whether it is advantageous to implement these 86:

managers can use Real Options Valuation to help them deal with various uncertainties in making decisions about the allocation of resources among R&D projects. Non-business examples might be evaluating the cost of

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As above, data issues arise as far as estimating key model inputs. Here, since the value or price of the underlying cannot be (directly) observed, there will always be some (much) uncertainty as to its value (i.e.

251:: Here the project is built with capacity in excess of the expected level of output so that it can produce at higher rates if needed. Management then has the option (but not the obligation) to expand – i.e. 1023:

Management must be in the position to exercise, in so far as some real options are proprietary (owned or exercisable by a single individual or a company) while others are shared (can (only) be exercised by many

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Management must understand options, be able to identify and create them, and appropriately exercise them. This contrasts with business leaders focused on maintaining the status quo and / or near-term accounting

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may be applied, as is the case with most option pricing models. (d) Under ROV however, the option and (usually) its underlying are clearly not traded, and forming a hedging portfolio would be difficult, if not

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It is often difficult to capture the rules relating to exercise, and consequent actions by management. Further, a project may have a portfolio of embedded real options, some of which may be mutually exclusive.

427:: Management may have the option to change the output rate per unit of time or to change the total length of production run time, for example in response to market conditions. These options are also known as 2135: 671:

generated by the underlying asset: As part of a project, the dividend equates to any income which could be derived from the real assets and paid to the owner. These reduce the appreciation of the asset.

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Corporate strategy has to be adaptive to contingent events. Some corporations face organizational rigidities and are unable to react to market changes; in this case, the NPV approach is appropriate.

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at each decision point, allowing for risk neutral valuation. Under ROV, however: (a) managements' actions actually change the risk characteristics of the project in question, and hence (b) the

514:. Even when employed, however, these latter methods do not normally properly account for changes in risk over the project's lifecycle and hence fail to appropriately adapt the risk adjustment. 306:

Where there is uncertainty as to when, and how, business or other conditions will eventuate, flexibility as to the timing of the relevant project(s) is valuable, and constitutes optionality.

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if it invests this year: 5M. If it invests next year, the discounted cash flows are 6M with a 66.7% probability, and 3M with a 33.3% probability. Assuming a risk neutral rate of 10%, future

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of the business must be such that it has the ability to fund the project as, and when, required (i.e. issue shares, absorb further debt and / or use internally generated cash flow); see

421:, for example, may have the option to switch between various fuel sources to produce electricity, and therefore a flexible plant, although more expensive may actually be more valuable. 4046: 960:

effectively during operations. These methods have been applied in many use cases in aerospace, defense, energy, transport, space, and water infrastructure design and planning.

682:: this corresponds to any (non-recoverable) investment outlays, typically the prospective costs of the project. In general, management would proceed (i.e. the option would be 1858: 578:

management's ability to respond to the evolution of these parameters. It is the combined effect of these that makes ROV technically more challenging than its alternatives.

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to make a sequel in the event the original movie is not successful. This real option has economic worth and can be valued monetarily using an option-pricing model. See

574:, ROV is distinguished from the latter, in that it takes into account uncertainty about the future evolution of the parameters that determine the value of the project, 313:: perhaps the most generic in this category – these entail the call option to exercise only those projects that appear to be profitable at the time of initiation. 269:: The project is engineered such that output can be contracted in future should conditions turn out to be unfavourable. Forgoing these future expenditures constitutes 351:. Here, when the present value of the remaining cash flows falls below the liquidation value, the asset may be sold, and this act is effectively the exercising of a 95:. It, thus, forces decision makers to be explicit about the assumptions underlying their projections, and for this reason ROV is increasingly employed as a tool in 3719: 144:

are, in present terms, 5.45M and 2.73M, respectively. The investment cost is 4M. If the firm invests next year, the present value of the investment cost is 3.63M.

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project; whereas this is not a consideration as regards the underlying security of a financial option. Moreover, management cannot measure uncertainty in terms of

3888: 490:. Here, only the expected cash flows are considered, and the "flexibility" to alter corporate strategy in view of actual market realizations is "ignored"; see 5397: 4039: 1397: 1238:

has been a leading name for many years, publishing several influential books and academic articles. Other pioneering academics in the field include Professors

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Whereas business managers have been making capital investment decisions for centuries, the term "real option" is relatively new, and was coined by Professor

696:: the time during which management may decide to act, or not act, corresponds to the life of the option. As above, examples include the time to expiry of a 5309: 2087: 1764:

Gunther McGrath, Rita; Nerkar, Atul (January 2004). "Real options reasoning and a new look at the R&D investment strategies of pharmaceutical firms".

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logic. Here, essentially: (a) it is presupposed that one can create a "hedged portfolio" comprising one option and "delta" shares of the underlying. (b)

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Standard option models: (a) Assume that the risk characteristics of the underlying do not change over the life of the option, usually expressed via a

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See Ch. 23, Sec. 5, in: Frank Reilly, Keith Brown (2011). "Investment Analysis and Portfolio Management." (10th Edition). South-Western College Pub.

396:. As in the preceding cases, this flexibility increases the value of the project, corresponding in turn, to the "premium" paid for the real option. 1715:"Public Funding for Science and the Value of Corporate R&D Projects; Evidence from Project Initiation and Termination Decisions in Cell Therapy" 3919: 2757: 2454: 1450: 722:. Management's ability to respond to changes in value is modeled at each decision point as a series of options, as above these may comprise, i.a.: 3948: 1878:"An evolutionary real options framework for the design and management of projects and systems with complex real options and exercising conditions" 2276: 1255: 517:

By contrast, ROV assumes that management is "active" and can "continuously" respond to market changes. Real options consider "all" scenarios (or

2847:"An approach for analyzing and managing flexibility in engineering systems design based on decision rules and multistage stochastic programming" 2175:

Comparing Theory With Reported Data for Reliability: Real Options Modeling of Actively Traded Philippine Universal Banks - SAGE Research Methods

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Practically, the business must be positioned such that it has appropriate information flow, and opportunities to act. This will often be a

871: 255: – should conditions turn out to be favourable. A project with the option to expand will cost more to establish, the excess being the 1133:, the data issues are usually addressed using a simulation of the project, or a listed proxy. Various new methods – see for example 1114: 1046:. Finally, even if the firm can actively adapt to market changes, it remains to determine the right paradigm to discount future claims 244:

Where the project's scope is uncertain, flexibility as to the size of the relevant facilities is valuable, and constitutes optionality.

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to find the optimal design and decision rule variables. A more recent approach reformulates the real option problem as a data-driven

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problems. Note that for American styled real options, this application is somewhat more complex; although recent research combines a

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This simple example shows the relevance of the real option to delay investment and wait for further information, and is adapted from

3114: 3967: 797:. The model must also be flexible enough to allow for the relevant decision rule to be coded appropriately at each decision point. 3861: 5443: 3559: 1254:(the latter two, authoring the pioneering text in the discipline). An academic conference on real options is organized yearly ( 5133: 1273:" for those businesses. Trigeorgis also has broadened exposure to real options through layman articles in publications such as 3529: 2556: 1223:). The description of such opportunities as "real options", however, followed on the development of analytical techniques for 859:

approach with simulation, allowing for the valuation of real options which are both multidimensional and American styled; see

3708: 3372: 3351: 3332: 3313: 3290: 3269: 3227: 3076: 2996:

Caputo, Cesare; Cardin, Michel-Alexandre; Ge, Pudong; Teng, Fei; Korre, Anna; Antonio del Rio Chanona, Ehecatl (2023-04-01).

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they can wait to decide on a sequel after the original film is released. The second approach, he states, provides the option

3958: 3822:"A Procedure for the Rapid Pre-acquisition Screening of Target Companies Using the Pay-off Method for Real Option Valuation" 3652:

Patent Damages and Real Options: How Judicial Characterization of Non-Infringing Alternatives Reduces Incentives to Innovate

534: 3884: 2941:"Analyzing Real Options and Flexibility in Engineering Systems Design Using Decision Rules and Deep Reinforcement Learning" 2687: 1678:

Oriani, Raffaele; Sobrero, Maurizio (2008). "Uncertainty and the market valuation of R&D within a real options logic".

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To use standard option pricing models here, despite the difficulties relating to rational pricing, practitioners adopt the

3906: 2066: 1915: 1470: 919:, which merge physical design considerations and management decisions through an intuitive "if-then-else" statement e.g., 5265: 3726: 3573:

Grenadier, Steven R. & Weiss, Allen M., 1997. "Investment in technological innovations: An option pricing approach,"

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To address the fact that changing characteristics invalidate the use of a constant discount rate, some analysts use the "

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once committed. Some analysts account for this uncertainty by (i) adjusting the discount rate, e.g. by increasing the

3391: 3246: 3165: 3130: 3457:. 5th Conference on Advances in Financial Technologies (AFT 2023). Schloss Dagstuhl – Leibniz-Zentrum für Informatik. 2894:"An approach based on robust optimization and decision rules for analyzing real options in engineering systems design" 615:

of the project is required: this is usually based on management's "best guess" as to the gross value of the project's

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Cardin, Michel-Alexandre; Mijic, Ana; Whyte, Jennifer (2020), Maier, Anja; Oehmen, Josef; Vermaas, Pieter E. (eds.),

2382: 1987: 510:, or (iii) applying (subjective) "haircuts" to the forecast numbers, or (iv) via probability-weighting these as in 180:

The firm does not know how well its stores are accepted in a foreign country. If their stores have high demand, the

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Schaum's quick guide to business formulas: 201 decision-making tools for business, finance, and accounting students

487: 393: 537:, while discounting at the risk-free rate. This technique is also known as the "martingale" approach, and uses a 1387: 1013: 229: 3745:"Stay in School or Start Working? - The Human Capital Investment Decision under Uncertainty and Irreversibility" 583:

values of the option at the various points where a decision is needed on whether to continue with the project...

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mining machines, or the decision to join the work force, or rather, to forgo several years of income to attend

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Datar, V.; Mathews, S. (2004). "European Real Options: An Intuitive Algorithm for the Black Scholes Formula".

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options or perpetual American options. Note that this application of Black–Scholes assumes constant — i.e.

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next year is 0.41M. Given this, the firm should opt by opening one store. This simple example shows that a

4090: 4012: 3987: 1173: 1094: 75: 5478: 5016: 4870: 4664: 4483: 3981: 2462: 1392: 952: 495: 479: 323:

exploration a firm can delay mining a deposit until market conditions are favorable. This constitutes an

3916: 2487: 2049: 794: 5600: 5595: 5590: 5334: 5275: 5097: 4446: 3945: 3520: 1407: 1372: 1269:. He uses real options to explain the gap between how the stock market prices some businesses and the " 1266: 1102: 955:

to evaluate a wide range of possible real option and design implementation strategies, well suited for

530: 2235: 5379: 5190: 4617: 4602: 4592: 4431: 4246: 2893: 2846: 2614: 1979: 1932: 1545: 1270: 1239: 932: 403:: The option to produce different outputs from the same facility is known as an output mix option or 2732: 5498: 5493: 4635: 4612: 4550: 4376: 4301: 4261: 4241: 3879: 3190: 2685:

Mathews, S.; Datar, V. (2007). "A Practical Method for Valuing Real Options: The Boeing Approach".

1215: 483: 347:: Management may have the option to cease a project during its life, and, possibly, to realise its 2036: 172: 5448: 5148: 5118: 5093: 4976: 4817: 4749: 4346: 4336: 4306: 4276: 4216: 3931: 2623: 2240: 1953: 1412: 1320:, in 1992, which may have been the first business school case study to teach ROV. Reflecting the 1305: 1274: 1110: 1086: 944: 837: 816:— costs: in cases where the project's costs, like its revenue, are also assumed stochastic, then 100: 3183: 1827: 1588: 259:, but is worth more than the same without the possibility of expansion. This is equivalent to a 5245: 5230: 5195: 4488: 4351: 4189: 4159: 3936: 3788: 3149: 2727: 2093: 1971: 936: 852: 801: 545:. For related discussion – and graphical representation – see 161:

may lead the firm to take unnecessary risk, which could be prevented by real options valuation.

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Cardin, Michel-Alexandre; Xie, Qihui; Ng, Tsan Sheng; Wang, Shuming; Hu, Junfei (2017-01-02).

2808:"Enabling Flexibility in Engineering Systems: A Taxonomy of Procedures and a Design Framework" 2322:

Tan, Jackson J. (2018-01-01). "Interfaces for enterprise valuation from a real options lens".

1951:

Timothy Luehrman: "Investment Opportunities as Real Options: Getting Started on the Numbers".

1220: 1089:. (c) When hedging of this sort is possible, since delta hedging and risk neutral pricing are 5585: 5458: 5225: 5123: 4802: 4503: 3901: 3659: 3490:

Brennan, Michael J.; Schwartz, Eduardo S. (1985). "Evaluating Natural Resource Investments".

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Limitations as to the use of these models arise due to the contrast between Real Options and

817: 656: 189: 185: 181: 152: 141: 137: 103:, because otherwise the complex compound real options will become too intractable to handle. 99:

formulation. This extension of real options to real-world projects often requires customized

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The valuation methods usually employed, likewise, are adapted from techniques developed for

498:. The NPV framework (implicitly) assumes that management is "passive" with regard to their 5412: 5369: 5359: 5349: 5344: 5070: 5011: 4946: 4900: 4895: 4769: 4729: 4696: 4525: 4520: 4024: 3897: 3421: 3305: 3019: 1603: 1503: 1321: 1297: 1078: 1059: 781: 693: 626: 558: 538: 507: 463: 124: 64: 38: 3736: 1574: 8: 5417: 5205: 5128: 4951: 4475: 3975: 3410:"Valuing the option to prototype: A case study with Generation Integrated Energy Storage" 2574:"The valuation of multidimensional American real options using the LSM simulation method" 1587:

Locatelli, Giorgio; Boarin, Sara; Pellegrino, Francesco; Ricotti, Marco E. (2015-02-01).

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Cortazar, Gonzalo (2000). "Simulation and Numerical Methods in Real Options Valuation".

2131:"A real options model for loan portfolios of actively traded Philippine universal banks" 1957:

76, no. 4 (July – August 1998): 51–67.; "Strategy as a Portfolio of Real Options".

1607: 1507: 5468: 5453: 5422: 5407: 5374: 5240: 5031: 4996: 4759: 4724: 4687: 4645: 4557: 4540: 4498: 4421: 4413: 4199: 4184: 4059: 3858: 3808: 3789:"Characteristics of Limited Entry Fisheries and the Option Component of Entry Licenses" 3775: 3744: 3742: 3679: 3507: 3478: 3458: 3280: 3258: 3179: 3009: 2978: 2921: 2874: 2640: 2479: 2402: 2197: 1897: 1537: 1332: 1262: 912: 726: 719: 660: 652: 499: 380: 357: 270: 252: 4582: 2998:"Design and planning of flexible mobile Micro-Grids using Deep Reinforcement Learning" 1998:

See: §32 "Certainty Equivalent Approach" & §165 "Risk Adjusted Discount Rate" in:

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per store is 5M. Assuming that the probability of both events is 50%, the expected

79: 35: 3161: 2909: 2862: 2272: 1231:

in 1973. As such, the term "real option" is closely tied to these option methods.

5564: 5534: 5529: 5483: 5319: 5314: 5260: 5170: 5078: 5051: 4991: 4986: 4956: 4905: 4890: 4807: 4787: 4597: 4508: 4461: 4451: 4386: 4311: 4281: 4231: 4179: 4151: 4095: 3971: 3952: 3923: 3910: 3865: 3763: 3434: 3409: 3134: 3056: 2560: 1615: 1559: 1515: 1357: 1144:, specific exercise rules can often be accommodated by coding these in a bespoke 1106: 956: 833: 809: 785: 751: 730: 503: 362: 324: 319:: Here management has flexibility as to when to start a project. For example, in 225: 92: 5539: 5524: 5324: 5235: 5185: 5162: 5143: 4971: 4913: 4880: 4875: 4855: 4779: 4545: 4456: 1251: 842: 822: 701: 636:; sometimes the volatility of the first period's cash flows are preferred; see 291:(FMS) is a good example of this type of option. This option is also known as a 278: 256: 221: 88: 3666:

Fernando Torres (2006-09-11). "Establishing Licensing Rates Through Options".

3473: 2592: 2398: 2149: 2130: 1893: 1062:(further complicated by uncertainty as to management's actions in the future). 5579: 5519: 5488: 5329: 5255: 5215: 5210: 5046: 4918: 4865: 4860: 4842: 4739: 4719: 4164: 4000: 3041: 2974: 2917: 2870: 2831: 2343: 2158: 1785: 1750: 1699: 1664: 1533: 1247: 1210: 1202: 1188:

discounting is required. (For an alternative, modifying Black-Scholes, see:.)

1161: 1117:

would be required, invalidating (technically) the risk neutrality assumption.

1037:, for which these were originally developed. The main difference is that the 998: 916: 901: 856: 813: 687: 629:: a measure for uncertainty as to the change in value over time is required: 471: 392:

Management may have flexibility relating to the product produced and /or the

348: 3138: 3127: 1020:. Management must, correspondingly, have appropriate access to this capital. 570:

Although there is much similarity between the modelling of real options and

470:(NPV). Under this "standard" NPV approach, future expected cash flows are 383:. Related here is also the notion of Intraproject vs. Interproject options. 5339: 5113: 5041: 5021: 4981: 4850: 4822: 4812: 4754: 4403: 4251: 4236: 4110: 3838: 3821: 2965: 2253: 1845: 1589:"Load following with Small Modular Reactors (SMR): A real options analysis" 1300:. The idea of treating strategic investments as options was popularized by 851:

have also been developed and are increasingly, and especially, applied to

845:(the exponent here, corresponding to the number of sources of uncertainty). 715: 683: 679: 518: 372: 2742: 2183: 644:

for a discussion relating to the estimation of NPV and project volatility.

5220: 5088: 5059: 5055: 5006: 4797: 4792: 4361: 4331: 4174: 4105: 3675: 3615:"Strategic Technology Investment Decisions in Research & Development" 3444: 1524: 1491: 1422: 1329: 754: 376: 327: 260: 3743:

Natasa Bilkic, Thomas Gries and Margarethe Pilichowski. (October 2012).

3364:

Real Options: Managerial Flexibility and Strategy in Resource Allocation

2600: 2362:

Pablo Fernandez (2019). "Valuing Real Options: Frequently Made Errors".

1999: 1624: 1398:

Government procurement in the United States § Real options analysis

1336:

choices. They can produce an original movie and sequel at the same time

1180:

be identical, by arbitrage arguments their values may (must) be equated

911:

By contrast, methods focusing on, for example, real option valuation in

70:

Real options analysis, as a discipline, extends from its application in

5544: 5180: 5175: 4941: 4827: 4607: 4567: 4371: 4321: 4291: 4169: 3812: 3771: 3582: 3511: 2644: 1731: 1714: 1234:

Real options are today an active field of academic research. Professor

1085:

arguments then allow for the option's price to be estimated today; see

1055: 1038: 733: 632:

the volatility in project value is generally used, usually derived via

608: 602: 478:

at a discount rate that reflects the embedded risk in the project; see

352: 274: 60: 2956: 2823: 1741: 861:

Monte Carlo methods for option pricing § Least Square Monte Carlo

808:-like solutions are sometimes employed. These are applicable only for 4734: 4656: 4341: 4204: 3601:"Using real options to make decisions in the motion picture industry" 2335: 1656: 1277:. This popularization is such that ROV is now a standard offering in 1082: 884: 883:, have been developed for real option valuation. These typically use 790: 784:. Note though that, in general, while most "real" problems allow for 616: 3871: 3804: 2940: 2807: 1777: 1691: 196:

Should the firm invest in one store, two stores, or not invest? The

5304: 5026: 4923: 4744: 4391: 4286: 4271: 3893: 3503: 3463: 3014: 2636: 668: 3650:

Sidak, J. Gregory; Leonard, Gregory K.; Hausman, Jerry A. (2007),

3600: 1490:

Locatelli, Giorgio; Mancini, Mauro; Lotti, Giovanni (2020-04-15).

533:. Here the approach, known as risk-neutral valuation, consists in 151:

rule for investment, the firm should invest this year because the

4221: 3219:

Real Options: Managing Strategic Investment in an Uncertain World

3128:

Does Risk-Neutral Valuation Mean that Investors Are Risk-Neutral?

1492:"A simple-to-implement real options method for the energy sector" 1213:

wrote explicitly of the "options" available to a business owner (

1125:

These issues are addressed via several interrelated assumptions:

1068:

Theoretical difficulties, which are more serious, may also arise.

908:, and the simulation with optimized exercise thresholds method. 443: 2000:

Joel G. Siegel; Jae K. Shim; Stephen Hartman (1 November 1997).

1586: 709: 5559: 3597: 3530:"Valuing Alternative Market Entry Strategies as "Real-Options"" 697: 337: 83: 3185:

Applications of Option-Pricing Theory: Twenty-Five Years Later

832:. These are more widely used given that most real options are 605:

is the project in question – it is modelled in terms of:

301: 4120: 3937:

Decision Making Under Uncertainty—Real Options to the Rescue?

1105:. (b) Hence a standard, risk free rate may be applied as the 915:

may be more sophisticated. These include analytics based on

535:

adjusting the probability distribution for risk consideration

4054: 3384:

Project valuation using real options: a practitioner's guide

3057:"Flexibility and Real Options in Engineering Systems Design" 3521:"Applications of option pricing theory to equity valuation" 3455:

Correct Cryptocurrency ASIC Pricing: Are Miners Overpaying?

2383:"Real Options Volatility Estimation with Correlated Inputs" 2050:

Capital Budgeting and Initial Cash Outlay (ICO) Uncertainty

1561:

Entrepreneurial Marketing; an effectual approach. Chapter 2

923:

demand is higher than a certain production capacity level,

708:. Note though that given the flexibility related to timing 511: 4021:, Prof. Steven T. Hackman, Georgia Institute of Technology 3965:

How Do You Assess The Value of A Company's "Real Options"?

3255: 3139:

Is It Possible to Use Real Options for Incomplete Markets?

3063:, Cham: Springer International Publishing, pp. 1–29, 2892:

Caunhye, Aakil M.; Cardin, Michel-Alexandre (2017-08-03).

2172:

Tan, Jackson J.; Trinidad, Fernando L. (January 3, 2019).

521:) and indicate the best corporate action in each of these 133:

economy, which can prevent it from investing with losses.

3982:

Some Important Issues Involving Real Options: An Overview

3583:

Don Chance; Eric Hillebrand; Jimmy Hilliard (June 2009).

2114:

Michael J. Mauboussin, Credit Suisse First Boston, 1999.

1282: 1113:

could differ depending on what state was realised, and a

978: 458:

ROV is often contrasted with more standard techniques of

2572:

Cortazar, Gonzalo; Gravet, Miguel; Urzua, Jorge (2008).

931:

do nothing; this approach can be combined with advanced

387: 2939:

Caputo, Cesare; Cardin, Michel-Alexandre (2021-09-21).

2082: 2080: 1643:(2017). "Real options theory in strategic management". 1452:

Real Options Analysis: Where are the Emperor's Clothes?

453: 184:

per store is 10M. If their stores have low demand, the

3988:

Real Power of Real Options, Leslie and Michaels (1997)

3720:"Real Options, Agency Conflicts, and Financial Policy" 3215: 2771:

de Neufville, Richard; Scholtes, Stefan (2011-08-12).

2229: 2227: 2225: 2223: 2221: 2219: 2217: 2048:

Michael C. Ehrhardt and John M. Wachowicz, Jr (2006).

1976:

Strategic Risk Taking: A Framework for Risk Management

1871: 1869: 1867: 1792: 1575:

Real Options Valuations: Taking Out the Rocket Science

341:

can shelve the patent and not incur any further costs.

3859:

Intro to Real Option Valuation as a Modelling Problem

3739:

G. Siller-Pagaza, G. Otalora, E. Cobas-Flores (2006).

3341: 2530: 2528: 2129:

Tan, Jackson J.; Trinidad, Fernando L. (2018-02-15).

1829:

Applications in Real Options and Value-based Strategy

1763: 1489: 3408:

Lai, Chun Sing; Locatelli, Giorgio (February 2021).

2770: 2720:

Journal of Applied Mathematics and Decision Sciences

2077: 1914:

This section draws primarily on Campbell R. Harvey:

3819: 3649: 3633:Heikkilä, Markku; Collan, Mikael (September 2011). 2621:(1985). "Evaluating Natural Resource Investments". 2571: 2271:Jenifer Piesse and Alexander Van de Putte. (2004). 2214: 1864: 1388:

Business valuation § Option pricing approaches

277:, and again, the excess upfront expenditure is the 239: 230:

Business valuation § Option pricing approaches

4015:, Prof. Aswath Damodaran, Stern School of Business 3599: 3584: 3257: 2995: 2525: 2357: 2355: 2353: 2001: 1312:Luehrman also co-authored with William Teichner a 491: 78:in general, adapting the techniques developed for 3717: 2713: 2117:Get Real: Using Real Options in Security Analysis 1328:discussed the essential points of Arundel in his 1261:Amongst others, the concept was "popularized" by 5577: 3553: 3381: 3256:Copeland, Thomas E.; Vladimir Antikarov (2001). 3154: 3054: 2716:"Fuzzy Pay-Off Method for Real Option Valuation" 2267: 2265: 2263: 1403:Principal–agent problem § Options framework 984: 892:(which can be understood as an extension of the 659:), or, if options exist on this security, their 506:, or (ii) adjusting the cash flows, e.g. using 3665: 3489: 3278: 3107: 2844: 2613: 2361: 2350: 2277:Annual International Conference on Real Options 1638: 1256:Annual International Conference on Real Options 211:does not imply that the firm should not invest. 3928:Investment Analysts Society of Southern Africa 3632: 2233: 2086:Lenos Trigeorgis, Rainer Brosch and Han Smit. 1992: 1383:Datar–Mathews method for real option valuation 1378:Fuzzy pay-off method for real option valuation 825:, typically valued using simulation as below.) 547:Datar–Mathews method for real option valuation 442:Given the above, it is clear that there is an 4672: 4040: 3737:The Impact of Real Options in Agency Problems 3160:Timothy A. Luehrman and William A. Teichner: 3113:Don M. Chance and Pamela P. Peterson (2002). 2891: 2260: 2110: 2108: 2106: 2104: 2102: 1677: 1292:Recently, real options have been employed in 866:When the Real Option can be modelled using a 224:. Real options are also commonly applied to 59:directly influence the value of the option's 3894:Valuing Real Options: Frequently Made Errors 3598:S. Young; J. Gong; W. Van der Stede (2012). 3579:, Elsevier, vol. 44(3), pages 397–416, June. 3527: 3518: 3407: 3344:Strategic Investment: Real Options and Games 3327:. Wiesbaden: Deutscher Universitäts-Verlag. 2938: 2756:: CS1 maint: DOI inactive as of July 2024 ( 2684: 2657: 2292: 2290: 2288: 2286: 2284: 1875: 1393:Corporate finance § Valuing flexibility 1265:, then chief U.S. investment strategist for 872:Finite difference methods for option pricing 795:Option (finance) § Model implementation 496:Corporate finance § Valuing flexibility 50:the alternative option to sell the factory. 3325:Real Option Valuation in Service Industries 3302:Real Options and Option-embedded Securities 2380: 2171: 2136:Journal of Global Entrepreneurship Research 2128: 1947: 1945: 1943: 1941: 1408:Patent valuation § Option-based method 1028: 690:of expected cash flows exceeds this amount; 302:Options relating to project life and timing 4679: 4665: 4641: 4631: 4047: 4033: 3452: 3360: 2714:Collan, M.; Fullér, R.; Mezei, J. (2009). 2448: 2446: 2099: 2029: 1964: 1961:76, no. 5 (September–October 1998): 87-99. 1927:This sub-section is additionally based on 1798: 565: 3946:Real Options Whitepapers and Case-studies 3837: 3612: 3472: 3462: 3443: 3433: 3346:. Princeton: Princeton University Press. 3285:. Princeton: Princeton University Press. 3222:. Boston: Harvard Business School Press. 3216:Amram, Martha; Kulatilaka, Nalin (1999). 3031: 3013: 2964: 2741: 2731: 2296: 2281: 2236:"A Real-World Way to Manage Real Options" 2148: 1840: 1838: 1822: 1820: 1740: 1730: 1712: 1623: 1523: 1441: 1439: 1437: 972: 541:. For technical considerations here, see 4006: 3556:"Real options in public infrastructures" 2553:Real Options with Monte Carlo Simulation 2426: 2420: 2059: 1938: 1465: 1463: 1461: 106: 5504:Power reverse dual-currency note (PRDC) 5444:Constant proportion portfolio insurance 3786: 3639:Journal of Intellectual Property Rights 3453:Yaish, Aviv; Zohar, Aviv (2020-06-18). 3162:"Arundel Partners: The Sequel Project." 2806:Cardin, Michel-Alexandre (2013-11-07). 2452: 2443: 2302:"The Promise and Peril of Real Options" 2273:"Volatility estimation in Real Options" 1296:, both for valuation purposes and as a 879:Various other methods, aimed mainly at 828:The most commonly employed methods are 110: 82:to "real-life" decisions. For example, 5578: 4686: 3984:, Gordon Sick and Andrea Gamba (2005). 3820:Collan, Mikael; Kinnunen Jani (2011). 3342:Smit, T.J.; Trigeorgis, Lenos (2004). 3322: 3061:Handbook of Engineering Systems Design 2805: 1851: 1835: 1817: 1577:, Strategic Finance, Feb. 2003, 10-13. 1552: 1434: 1134: 4660: 4028: 3939:, Prof. Luke Miller & Chan Park, 3872:The Promise and Peril of Real Options 3299: 3236: 3120: 3115:Real Options and Investment Valuation 2510:See pg 26 in Marion A. Brach (2003). 1458: 1418:Present value of growth opportunities 1087:Rational pricing § Delta hedging 655:, using either its price volatility ( 388:Options relating to project operation 5439:Collateralized debt obligation (CDO) 4001:Journal of Real Options and Strategy 3826:Journal of Real Options and Strategy 3698: 3260:Real Options: A Practitioner's Guide 3090: 2688:Journal of Applied Corporate Finance 2544: 2068:Calculating value during uncertainty 1485: 1483: 1481: 1318:Arundel Partners: The Sequel Project 904:and economic decision-making), the 775: 767:which may also apply to the project. 454:Applicability of standard techniques 233: 3699:Ronn, Ehud I.; et al. (2002). 3382:Kodukula, P.; Papudesu, C. (2006). 2581:Computers & Operations Research 2381:Cobb, Barry; Charnes, John (2004). 2321: 1859:Real Options: Opportunity from Risk 1713:Huang, Hsini; Jong, Simcha (2019). 1077:Option pricing models are built on 743:the project (also an American put); 591: 13: 3701:Real Options and Energy Management 3204: 3166:Harvard Business School Publishing 1573:Amram, M., and K. N. Howe (2003), 1314:Harvard Business School case study 1137: – also address these issues. 836:. Additionally, and particularly, 793:between these considerations; see 334:Delay option with a product patent 14: 5612: 3847: 3586:"Pricing options on film revenue" 3198:, vol. 88(3), pages 323–49, June. 3117:. The Research Foundation of AIMR 2774:Flexibility in Engineering Design 2455:"An Introduction to Real Options" 2234:Copeland, T.; Tufano, P. (2004). 1876:Zhang, S.X.; Babovic, V. (2011). 1811: 1478: 355:. This option is also known as a 76:decision making under uncertainty 5558: 4640: 4630: 3959:Real Options – Introduction 2701:10.1111/j.1745-6622.2007.00140.x 2073:IBM Institute for Business Value 1933:The Option to Expand and Abandon 240:Options relating to project size 3917:An introduction to real options 3172: 3143: 3048: 2989: 2932: 2885: 2838: 2799: 2764: 2707: 2678: 2651: 2607: 2565: 2504: 2374: 2315: 2165: 2122: 2042: 1921: 1908: 1846:Thinking in Real (Options) Time 1804: 1757: 849:Specialised Monte Carlo Methods 712:, caution must be applied here. 317:Initiation or deferment options 234:"Applications" referenced below 16:Capital budgeting analysis term 5266:Year-on-year inflation-indexed 4427:Debtor-in-possession financing 3576:Journal of Financial Economics 3279:Dixit, A.; R. Pindyck (1994). 3098:Reality Check for Real Options 3069:10.1007/978-3-030-46054-9_35-1 3033:10.1016/j.apenergy.2023.120707 2783:10.7551/mitpress/8292.001.0001 2476:10.1080/10293523.2004.11082463 1706: 1671: 1632: 1580: 1567: 1368:Monte Carlo methods in finance 1207:MIT Sloan School of Management 1170:replicating portfolio approach 1103:constant volatility assumption 963: 273:. This is the equivalent to a 232:- as well as to various other 1: 5276:Zero-coupon inflation-indexed 3787:Karpoff, Jonathan M. (1989). 3196:American Economic Association 3188:, Pages 107, 115; reprinted: 2910:10.1080/24725854.2017.1299958 2863:10.1080/0740817X.2016.1189627 1719:Journal of Management Studies 1428: 985:Organizational considerations 868:partial differential equation 476:empirical probability measure 413:: An input mix option – 289:flexible manufacturing system 4367:Staggered board of directors 3909:, Prof. Campbell R. Harvey. 3764:10.1016/j.labeco.2012.04.005 3718:D. Mauer; S. Sarkar (2001). 3635:"Enhancing Patent Valuation" 3435:10.1016/j.energy.2020.119290 3367:. Cambridge: The MIT Press. 3282:Investment Under Uncertainty 2945:Journal of Mechanical Design 2812:Journal of Mechanical Design 2008:. McGraw-Hill Professional. 1832:; Ch.4. in Trigeorgis (1996) 1766:Strategic Management Journal 1680:Strategic Management Journal 1645:Strategic Management Journal 1616:10.1016/j.energy.2014.11.040 1516:10.1016/j.energy.2020.117226 1279:postgraduate finance degrees 1049:The difficulties, are then: 1018:Financial statement analysis 750:or extend the project (both 559:volatility of the underlying 437: 285:Option to expand or contract 7: 5479:Foreign exchange derivative 4871:Callable bull/bear contract 4484:Accretion/dilution analysis 3994: 2463:Investment Analysts Journal 1351: 953:deep reinforcement learning 647:some analysts substitute a 394:process used in manufacture 173:"Staged Investment Example" 10: 5619: 4447:Leveraged recapitalization 3913:, Fuqua School of Business 3361:Trigeorgis, Lenos (1996). 3300:Moore, William T. (2001). 2660:Journal of Applied Finance 1373:Contingent claim valuation 1267:Credit Suisse First Boston 1196: 927:expand existing capacity, 611:: the starting or current 531:contingent claims analysis 361:. Abandonment options are 115: 5553: 5512: 5431: 5388: 5380:Stock market index future 5284: 5161: 5069: 4932: 4841: 4778: 4712: 4703: 4694: 4626: 4618:Valuation using multiples 4603:Sum-of-the-parts analysis 4573:Modigliani–Miller theorem 4474: 4432:Dividend recapitalization 4412: 4260: 4247:Secondary market offering 4150: 4139: 4066: 3974:, Prof. Alfred Rappaport 3896:, Prof. Pablo Fernandez, 3852: 3474:10.4230/LIPIcs.AFT.2023.2 3237:Brach, Marion A. (2003). 3169:case no. 9-292-140 (1992) 2593:10.1016/j.cor.2006.02.016 2399:10.1080/00137910490453392 2387:The Engineering Economist 2150:10.1186/s40497-018-0091-9 2037:Valuing Firms in Distress 1980:Wharton School Publishing 1894:10.1016/j.dss.2010.12.001 1001:and / or a firm enjoying 933:mathematical optimization 782:valuing financial options 446:between real options and 5499:Mortgage-backed security 5494:Interest rate derivative 5469:Equity-linked note (ELN) 5454:Credit-linked note (CLN) 4636:List of investment banks 4551:Free cash flow to equity 4377:Super-majority amendment 4302:Management due diligence 4242:Seasoned equity offering 3907:Identifying real options 3880:Stern School of Business 3239:Real Options in Practice 3191:American Economic Review 3150:valuebasedmanagement.net 2512:Real Options in Practice 1916:Identifying Real Options 1882:Decision Support Systems 1848:, businessfinancemag.com 1810:See Bilkic et al. under 1475:, Duke University, 2002. 1472:Identifying real options 1029:Technical considerations 676:Option characteristics: 101:decision support systems 53: 5449:Contract for difference 4750:Risk-free interest rate 4347:Shareholder rights plan 4337:Post-merger integration 4307:Managerial entrenchment 4277:Contingent value rights 4217:Initial public offering 4019:Real Options Calculator 3932:University of Cape Town 3668:SSRN Electronic Journal 3492:The Journal of Business 3323:Müller, Jürgen (2000). 2624:The Journal of Business 2241:Harvard Business Review 2094:Dow Jones & Company 1959:Harvard Business Review 1954:Harvard Business Review 1413:Contingent value rights 1275:The Wall Street Journal 1151:The theoretical issues: 1111:Required rate of return 945:Markov decision process 900:with an adjustment for 566:Options based valuation 425:Operating scale options 5231:Forward Rate Agreement 4489:Adjusted present value 4352:Special-purpose entity 4190:Direct public offering 4160:At-the-market offering 4013:ROV Spreadsheet Models 3978:and Michael Mauboussin 3839:10.12949/realopn.4.117 3703:. London: Risk Books. 3554:Richard de Neufville. 3386:. J. Ross Publishing. 2746:(inactive 2024-07-29). 1861:. archived 2010-04-05. 1799:Yaish & Zohar 2020 1216:The Theory of Interest 1174:Risk neutral valuation 1115:premium over risk free 1095:risk neutral valuation 973:Market characteristics 937:stochastic programming 634:monte carlo simulation 585: 20:Real options valuation 5459:Credit default option 4803:Employee stock option 4504:Conglomerate discount 4007:Calculation resources 3902:University of Navarra 3885:Real Options Tutorial 3306:John Wiley & Sons 2184:10.4135/9781526479952 1857:David Shimko (2009). 1469:Campbell, R. Harvey. 1281:, and often, even in 1263:Michael J. Mauboussin 1135:those described above 657:historical volatility 580: 529:in the literature on 508:certainty equivalents 190:discounted cash flows 186:discounted cash flows 182:discounted cash flows 153:discounted cash flows 142:discounted cash flows 138:discounted cash flows 107:Types of real options 24:real options analysis 5413:Inflation derivative 5398:Commodity derivative 5370:Single-stock futures 5360:Normal backwardation 5350:Interest rate future 5191:Conditional variance 4697:Derivative (finance) 4526:Economic value added 4521:Discounted cash flow 3930:), Prof E. Gilbert, 3898:IESE Business School 3887:, Prof. Marco Dias, 3676:10.2139/ssrn.1014743 3264:. New York: Texere. 1558:Nijssen, E. (2014). 1298:conceptual framework 947:, and uses advanced 906:fuzzy pay-off method 890:Datar–Mathews method 838:lattice-based models 704:for a new mine. See 539:risk-neutral measure 464:discounted cash flow 125:"Investment Example" 39:valuation techniques 22:, also often termed 5565:Business portal 5418:Property derivative 4111:Senior secured debt 3976:Columbia University 3426:2021Ene...21719290L 3241:. New York: Wiley. 3024:2023ApEn..33520707C 2743:10.1155/2009/238196 2453:Gilbert, E (2004). 2054:Financial Decisions 2035:Aswath Damodaran: 1972:Risk Adjusted Value 1639:Trigeorgis, Lenos; 1608:2015Ene....80...41L 1564:, Routelegde, 2014. 1508:2020Ene...19717226L 1447:Stanford University 1346:Option (filmmaking) 941:robust optimisation 415:process flexibility 405:product flexibility 267:Option to contract 253:exercise the option 136:The firm knows its 113: 5423:Weather derivative 5408:Freight derivative 5390:Exotic derivatives 5310:Commodities future 4997:Intermarket spread 4760:Synthetic position 4688:Derivatives market 4646:Outline of finance 4558:Market value added 4541:Financial modeling 4499:Business valuation 4422:Debt restructuring 4200:Follow-on offering 4185:Corporate spin-off 4143:(terms/conditions) 4060:investment banking 3970:2019-10-20 at the 3961:, Portfolion Group 3955:, Dr. Jonathan Mun 3951:2012-02-08 at the 3922:2012-05-25 at the 3864:2012-11-15 at the 3528:Daryl G. Waldron. 3519:Aswath Damodaran. 3133:2010-07-16 at the 2559:2010-03-18 at the 1970:Aswath Damodaran: 1732:10.1111/joms.12423 1324:of ROV, Professor 1285:curricula at many 1209:in 1977. In 1930, 1014:financial position 1003:economies of scale 913:engineering design 818:Margrabe's formula 727:option to contract 661:implied volatility 500:Capital Investment 401:Output mix options 381:corporate strategy 369:Sequencing options 358:Termination option 111: 5601:Capital budgeting 5596:Options (finance) 5591:Corporate finance 5573: 5572: 5474:Equity derivative 5464:Credit derivative 5432:Other derivatives 5403:Energy derivative 5365:Perpetual futures 5246:Overnight indexed 5196:Constant maturity 5157: 5156: 5104:Finite difference 5037:Protective option 4654: 4653: 4578:Net present value 4563:Minority interest 4494:Associate company 4470: 4469: 4437:Financial sponsor 4357:Special situation 4327:Pre-emption right 4317:Minority discount 4227:Private placement 4126:Subordinated debt 4081:Exchangeable debt 4068:Capital structure 4056:Corporate finance 3941:Auburn University 3710:978-1-899332-98-4 3606:Strategic Finance 3374:978-0-262-20102-5 3353:978-0-691-01039-7 3334:978-3-8244-7138-6 3315:978-0-471-21659-9 3292:978-0-691-03410-2 3271:978-1-58799-028-1 3229:978-0-87584-845-7 3104:, September, 2001 3078:978-3-030-46054-9 2957:10.1115/1.4052299 2898:IISE Transactions 2851:IISE Transactions 2824:10.1115/1.4025704 2792:978-0-262-30356-9 2777:. The MIT Press. 2309:NYU Working Paper 2298:Damodaran, Aswath 2193:978-1-5264-7995-2 2092:, copyright 2009 2015:978-0-07-058031-2 1641:Reuer, Jeffrey J. 1294:business strategy 1225:financial options 1172:", as opposed to 1035:financial options 898:Monte Carlo model 894:net present value 830:binomial lattices 776:Valuation methods 765:composite options 761:switching options 741:option to abandon 706:Option time value 686:) given that the 642:Corporate finance 572:financial options 555:financial options 527:financial options 523:contingent events 468:net present value 466:(DCF) analysis / 460:capital budgeting 448:financial options 430:Intensity options 411:Input mix options 345:Option to abandon 217: 216: 209:net present value 198:net present value 166:Staged Investment 159:net present value 149:net present value 97:business strategy 80:financial options 72:corporate finance 43:capital budgeting 5608: 5563: 5562: 5335:Forwards pricing 5109:Garman–Kohlhagen 4710: 4709: 4681: 4674: 4667: 4658: 4657: 4644: 4643: 4634: 4633: 4536:Fairness opinion 4531:Enterprise value 4514:Weighted average 4442:Leveraged buyout 4297:Drag-along right 4195:Equity carve-out 4152:Equity offerings 4148: 4147: 4144: 4116:Shareholder loan 4101:Second lien debt 4096:Preferred equity 4076:Convertible debt 4049: 4042: 4035: 4026: 4025: 3876:Aswath Damodaran 3843: 3841: 3816: 3783: 3752:Labour Economics 3749: 3733: 3731: 3725:. Archived from 3724: 3714: 3695: 3662: 3646: 3629: 3627: 3626: 3617:. Archived from 3609: 3603: 3594: 3588: 3570: 3568: 3567: 3558:. Archived from 3550: 3548: 3547: 3541: 3535:. Archived from 3534: 3524: 3515: 3486: 3476: 3466: 3449: 3447: 3437: 3397: 3378: 3357: 3338: 3319: 3296: 3275: 3263: 3252: 3233: 3199: 3176: 3170: 3158: 3152: 3147: 3141: 3126:See Marco Dias: 3124: 3118: 3111: 3105: 3094: 3088: 3087: 3086: 3085: 3052: 3046: 3045: 3035: 3017: 2993: 2987: 2986: 2968: 2936: 2930: 2929: 2889: 2883: 2882: 2842: 2836: 2835: 2803: 2797: 2796: 2768: 2762: 2761: 2755: 2747: 2745: 2735: 2711: 2705: 2704: 2682: 2676: 2675: 2655: 2649: 2648: 2611: 2605: 2604: 2578: 2569: 2563: 2548: 2542: 2532: 2523: 2508: 2502: 2501: 2499: 2498: 2492: 2486:. Archived from 2459: 2450: 2441: 2440: 2429:EFMA 2000 Athens 2424: 2418: 2417: 2415: 2413: 2378: 2372: 2371: 2359: 2348: 2347: 2336:10.1002/jsc.2181 2324:Strategic Change 2319: 2313: 2312: 2306: 2294: 2279: 2269: 2258: 2257: 2248:(3): 90–9, 128. 2231: 2212: 2211: 2209: 2208: 2169: 2163: 2162: 2152: 2126: 2120: 2112: 2097: 2084: 2075: 2063: 2057: 2046: 2040: 2033: 2027: 2026: 2024: 2022: 2007: 1996: 1990: 1968: 1962: 1949: 1936: 1929:Aswath Damodaran 1925: 1919: 1912: 1906: 1905: 1873: 1862: 1855: 1849: 1842: 1833: 1824: 1815: 1808: 1802: 1796: 1790: 1789: 1761: 1755: 1754: 1744: 1734: 1725:(5): 1000–1039. 1710: 1704: 1703: 1675: 1669: 1668: 1657:10.1002/smj.2593 1636: 1630: 1629: 1627: 1593: 1584: 1578: 1571: 1565: 1556: 1550: 1549: 1544:. Archived from 1527: 1487: 1476: 1467: 1456: 1443: 1363:Opportunity cost 1326:Robert C. Merton 1302:Timothy Luehrman 1287:Business Schools 1244:Eduardo Schwartz 1236:Lenos Trigeorgis 1093:identical, then 1079:rational pricing 1044:market liquidity 949:machine learning 853:high-dimensional 748:option to expand 729:the project (an 592:Valuation inputs 419:electric utility 336:: A firm with a 321:natural resource 294:Switching option 249:Option to expand 176: 128: 114: 112:Simple Examples 5618: 5617: 5611: 5610: 5609: 5607: 5606: 5605: 5576: 5575: 5574: 5569: 5557: 5549: 5535:Great Recession 5530:Government debt 5508: 5484:Fund derivative 5427: 5384: 5345:Futures pricing 5320:Dividend future 5315:Currency future 5298: 5280: 5153: 5129:Put–call parity 5065: 5052:Vertical spread 4987:Diagonal spread 4957:Calendar spread 4928: 4837: 4774: 4699: 4690: 4685: 4655: 4650: 4622: 4598:Stock valuation 4593:Residual income 4509:Cost of capital 4466: 4462:Project finance 4452:High-yield debt 4408: 4387:Tag-along right 4312:Mandatory offer 4282:Control premium 4263: 4256: 4232:Public offering 4180:Bought out deal 4142: 4141: 4135: 4062: 4053: 4009: 3997: 3972:Wayback Machine 3953:Wayback Machine 3924:Wayback Machine 3911:Duke University 3868:, Mikael Collan 3866:Wayback Machine 3855: 3850: 3805:10.2307/3146806 3747: 3729: 3722: 3711: 3624: 3622: 3613:David Lackner. 3565: 3563: 3545: 3543: 3539: 3532: 3394: 3375: 3354: 3335: 3316: 3293: 3272: 3249: 3230: 3210:Standard texts: 3207: 3205:Further reading 3202: 3178:Robert Merton, 3177: 3173: 3159: 3155: 3148: 3144: 3135:Wayback Machine 3125: 3121: 3112: 3108: 3095: 3091: 3083: 3081: 3079: 3053: 3049: 2994: 2990: 2937: 2933: 2890: 2886: 2843: 2839: 2804: 2800: 2793: 2769: 2765: 2749: 2748: 2733:10.1.1.534.2962 2726:(13601): 1–15. 2712: 2708: 2683: 2679: 2656: 2652: 2612: 2608: 2576: 2570: 2566: 2561:Wayback Machine 2549: 2545: 2533: 2526: 2509: 2505: 2496: 2494: 2490: 2457: 2451: 2444: 2425: 2421: 2411: 2409: 2379: 2375: 2360: 2351: 2320: 2316: 2304: 2295: 2282: 2270: 2261: 2232: 2215: 2206: 2204: 2194: 2170: 2166: 2127: 2123: 2113: 2100: 2085: 2078: 2064: 2060: 2047: 2043: 2034: 2030: 2020: 2018: 2016: 1997: 1993: 1969: 1965: 1950: 1939: 1926: 1922: 1913: 1909: 1874: 1865: 1856: 1852: 1844:Joanne Sammer: 1843: 1836: 1826:Justin Pettit: 1825: 1818: 1809: 1805: 1797: 1793: 1778:10.1002/smj.358 1762: 1758: 1711: 1707: 1692:10.1002/smj.664 1676: 1672: 1637: 1633: 1591: 1585: 1581: 1572: 1568: 1557: 1553: 1488: 1479: 1468: 1459: 1444: 1435: 1431: 1358:Option contract 1354: 1322:"mainstreaming" 1271:intrinsic value 1240:Michael Brennan 1199: 1031: 987: 975: 966: 957:complex systems 896:multi-scenario 834:American styled 810:European styled 778: 752:American styled 731:American styled 720:option exercise 649:listed security 594: 568: 504:cost of capital 456: 440: 390: 363:American styled 325:American styled 304: 271:option exercise 242: 226:stock valuation 171: 168: 164: 163: 123: 120: 109: 93:graduate school 56: 17: 12: 11: 5: 5616: 5615: 5604: 5603: 5598: 5593: 5588: 5571: 5570: 5568: 5567: 5554: 5551: 5550: 5548: 5547: 5542: 5540:Municipal debt 5537: 5532: 5527: 5525:Corporate debt 5522: 5516: 5514: 5510: 5509: 5507: 5506: 5501: 5496: 5491: 5486: 5481: 5476: 5471: 5466: 5461: 5456: 5451: 5446: 5441: 5435: 5433: 5429: 5428: 5426: 5425: 5420: 5415: 5410: 5405: 5400: 5394: 5392: 5386: 5385: 5383: 5382: 5377: 5372: 5367: 5362: 5357: 5352: 5347: 5342: 5337: 5332: 5327: 5325:Forward market 5322: 5317: 5312: 5307: 5301: 5299: 5297: 5296: 5291: 5285: 5282: 5281: 5279: 5278: 5273: 5268: 5263: 5258: 5253: 5248: 5243: 5238: 5233: 5228: 5223: 5218: 5213: 5208: 5206:Credit default 5203: 5198: 5193: 5188: 5183: 5178: 5173: 5167: 5165: 5159: 5158: 5155: 5154: 5152: 5151: 5146: 5141: 5136: 5131: 5126: 5121: 5116: 5111: 5106: 5101: 5091: 5086: 5081: 5075: 5073: 5067: 5066: 5064: 5063: 5049: 5044: 5039: 5034: 5029: 5024: 5019: 5014: 5009: 5004: 5002:Iron butterfly 4999: 4994: 4989: 4984: 4979: 4974: 4972:Covered option 4969: 4964: 4959: 4954: 4949: 4944: 4938: 4936: 4930: 4929: 4927: 4926: 4921: 4916: 4911: 4910:Mountain range 4908: 4903: 4898: 4893: 4888: 4883: 4878: 4873: 4868: 4863: 4858: 4853: 4847: 4845: 4839: 4838: 4836: 4835: 4830: 4825: 4820: 4815: 4810: 4805: 4800: 4795: 4790: 4784: 4782: 4776: 4775: 4773: 4772: 4767: 4762: 4757: 4752: 4747: 4742: 4737: 4732: 4727: 4722: 4716: 4714: 4707: 4701: 4700: 4695: 4692: 4691: 4684: 4683: 4676: 4669: 4661: 4652: 4651: 4649: 4648: 4638: 4627: 4624: 4623: 4621: 4620: 4615: 4613:Terminal value 4610: 4605: 4600: 4595: 4590: 4585: 4580: 4575: 4570: 4565: 4560: 4555: 4554: 4553: 4546:Free cash flow 4543: 4538: 4533: 4528: 4523: 4518: 4517: 4516: 4506: 4501: 4496: 4491: 4486: 4480: 4478: 4472: 4471: 4468: 4467: 4465: 4464: 4459: 4457:Private equity 4454: 4449: 4444: 4439: 4434: 4429: 4424: 4418: 4416: 4410: 4409: 4407: 4406: 4401: 4400: 4399: 4389: 4384: 4379: 4374: 4369: 4364: 4359: 4354: 4349: 4344: 4339: 4334: 4329: 4324: 4319: 4314: 4309: 4304: 4299: 4294: 4289: 4284: 4279: 4274: 4268: 4266: 4258: 4257: 4255: 4254: 4249: 4244: 4239: 4234: 4229: 4224: 4219: 4214: 4213: 4212: 4202: 4197: 4192: 4187: 4182: 4177: 4172: 4167: 4162: 4156: 4154: 4145: 4137: 4136: 4134: 4133: 4128: 4123: 4118: 4113: 4108: 4103: 4098: 4093: 4088: 4086:Mezzanine debt 4083: 4078: 4072: 4070: 4064: 4063: 4052: 4051: 4044: 4037: 4029: 4023: 4022: 4016: 4008: 4005: 4004: 4003: 3996: 3993: 3992: 3991: 3985: 3979: 3962: 3956: 3943: 3934: 3914: 3904: 3891: 3882: 3869: 3854: 3851: 3849: 3848:External links 3846: 3845: 3844: 3832:(1): 117–141. 3817: 3799:(4): 386–393. 3793:Land Economics 3784: 3758:(5): 706–717. 3740: 3734: 3732:on 2016-03-04. 3715: 3709: 3696: 3663: 3647: 3630: 3610: 3595: 3580: 3571: 3551: 3525: 3516: 3504:10.1086/296288 3498:(2): 135–157. 3487: 3450: 3399: 3398: 3393:978-1932159431 3392: 3379: 3373: 3358: 3352: 3339: 3333: 3320: 3314: 3297: 3291: 3276: 3270: 3253: 3248:978-0471445562 3247: 3234: 3228: 3206: 3203: 3201: 3200: 3171: 3153: 3142: 3119: 3106: 3089: 3077: 3047: 3002:Applied Energy 2988: 2931: 2904:(8): 753–767. 2884: 2837: 2798: 2791: 2763: 2706: 2677: 2650: 2637:10.1086/296288 2631:(2): 135–157. 2606: 2564: 2543: 2524: 2503: 2442: 2419: 2393:(2): 119–137. 2373: 2349: 2314: 2311:(S-DRP-05-02). 2280: 2259: 2213: 2192: 2164: 2121: 2098: 2076: 2065:Dan Latimore: 2058: 2056:, Summer 2006. 2041: 2028: 2014: 1991: 1963: 1937: 1920: 1907: 1888:(1): 119–129. 1863: 1850: 1834: 1816: 1803: 1791: 1756: 1705: 1686:(4): 343–361. 1670: 1631: 1579: 1566: 1551: 1548:on 4 May 2020. 1477: 1457: 1445:Adam Borison ( 1432: 1430: 1427: 1426: 1425: 1420: 1415: 1410: 1405: 1400: 1395: 1390: 1385: 1380: 1375: 1370: 1365: 1360: 1353: 1350: 1252:Robert Pindyck 1198: 1195: 1194: 1193: 1192: 1191: 1190: 1189: 1166: 1153: 1152: 1149: 1138: 1123: 1122: 1121: 1120: 1119: 1118: 1099: 1091:mathematically 1070: 1069: 1066: 1063: 1030: 1027: 1026: 1025: 1021: 1010: 1006: 995: 986: 983: 974: 971: 965: 962: 917:decision rules 877: 876: 864: 846: 826: 823:rainbow option 786:American style 777: 774: 773: 772: 771: 770: 769: 768: 758: 744: 737: 713: 702:mineral rights 691: 674: 673: 672: 666: 665: 664: 645: 624: 619:and resultant 593: 590: 567: 564: 472:present valued 455: 452: 439: 436: 435: 434: 422: 408: 389: 386: 385: 384: 366: 342: 331: 314: 311:Growth options 303: 300: 299: 298: 282: 279:option premium 264: 257:option premium 241: 238: 222:option premium 215: 214: 147:Following the 108: 105: 89:cryptocurrency 55: 52: 15: 9: 6: 4: 3: 2: 5614: 5613: 5602: 5599: 5597: 5594: 5592: 5589: 5587: 5584: 5583: 5581: 5566: 5561: 5556: 5555: 5552: 5546: 5543: 5541: 5538: 5536: 5533: 5531: 5528: 5526: 5523: 5521: 5520:Consumer debt 5518: 5517: 5515: 5513:Market issues 5511: 5505: 5502: 5500: 5497: 5495: 5492: 5490: 5489:Fund of funds 5487: 5485: 5482: 5480: 5477: 5475: 5472: 5470: 5467: 5465: 5462: 5460: 5457: 5455: 5452: 5450: 5447: 5445: 5442: 5440: 5437: 5436: 5434: 5430: 5424: 5421: 5419: 5416: 5414: 5411: 5409: 5406: 5404: 5401: 5399: 5396: 5395: 5393: 5391: 5387: 5381: 5378: 5376: 5373: 5371: 5368: 5366: 5363: 5361: 5358: 5356: 5353: 5351: 5348: 5346: 5343: 5341: 5338: 5336: 5333: 5331: 5330:Forward price 5328: 5326: 5323: 5321: 5318: 5316: 5313: 5311: 5308: 5306: 5303: 5302: 5300: 5295: 5292: 5290: 5287: 5286: 5283: 5277: 5274: 5272: 5269: 5267: 5264: 5262: 5259: 5257: 5254: 5252: 5249: 5247: 5244: 5242: 5241:Interest rate 5239: 5237: 5234: 5232: 5229: 5227: 5224: 5222: 5219: 5217: 5214: 5212: 5209: 5207: 5204: 5202: 5199: 5197: 5194: 5192: 5189: 5187: 5184: 5182: 5179: 5177: 5174: 5172: 5169: 5168: 5166: 5164: 5160: 5150: 5147: 5145: 5142: 5140: 5137: 5135: 5134:MC Simulation 5132: 5130: 5127: 5125: 5122: 5120: 5117: 5115: 5112: 5110: 5107: 5105: 5102: 5099: 5095: 5094:Black–Scholes 5092: 5090: 5087: 5085: 5082: 5080: 5077: 5076: 5074: 5072: 5068: 5061: 5057: 5053: 5050: 5048: 5047:Risk reversal 5045: 5043: 5040: 5038: 5035: 5033: 5030: 5028: 5025: 5023: 5020: 5018: 5015: 5013: 5010: 5008: 5005: 5003: 5000: 4998: 4995: 4993: 4990: 4988: 4985: 4983: 4980: 4978: 4977:Credit spread 4975: 4973: 4970: 4968: 4965: 4963: 4960: 4958: 4955: 4953: 4950: 4948: 4945: 4943: 4940: 4939: 4937: 4935: 4931: 4925: 4922: 4920: 4917: 4915: 4912: 4909: 4907: 4904: 4902: 4901:Interest rate 4899: 4897: 4896:Forward start 4894: 4892: 4889: 4887: 4884: 4882: 4879: 4877: 4874: 4872: 4869: 4867: 4864: 4862: 4859: 4857: 4854: 4852: 4849: 4848: 4846: 4844: 4840: 4834: 4831: 4829: 4826: 4824: 4823:Option styles 4821: 4819: 4816: 4814: 4811: 4809: 4806: 4804: 4801: 4799: 4796: 4794: 4791: 4789: 4786: 4785: 4783: 4781: 4777: 4771: 4768: 4766: 4763: 4761: 4758: 4756: 4753: 4751: 4748: 4746: 4743: 4741: 4740:Open interest 4738: 4736: 4733: 4731: 4728: 4726: 4723: 4721: 4720:Delta neutral 4718: 4717: 4715: 4711: 4708: 4706: 4702: 4698: 4693: 4689: 4682: 4677: 4675: 4670: 4668: 4663: 4662: 4659: 4647: 4639: 4637: 4629: 4628: 4625: 4619: 4616: 4614: 4611: 4609: 4606: 4604: 4601: 4599: 4596: 4594: 4591: 4589: 4586: 4584: 4581: 4579: 4576: 4574: 4571: 4569: 4566: 4564: 4561: 4559: 4556: 4552: 4549: 4548: 4547: 4544: 4542: 4539: 4537: 4534: 4532: 4529: 4527: 4524: 4522: 4519: 4515: 4512: 4511: 4510: 4507: 4505: 4502: 4500: 4497: 4495: 4492: 4490: 4487: 4485: 4482: 4481: 4479: 4477: 4473: 4463: 4460: 4458: 4455: 4453: 4450: 4448: 4445: 4443: 4440: 4438: 4435: 4433: 4430: 4428: 4425: 4423: 4420: 4419: 4417: 4415: 4411: 4405: 4402: 4398: 4395: 4394: 4393: 4390: 4388: 4385: 4383: 4380: 4378: 4375: 4373: 4370: 4368: 4365: 4363: 4360: 4358: 4355: 4353: 4350: 4348: 4345: 4343: 4340: 4338: 4335: 4333: 4330: 4328: 4325: 4323: 4320: 4318: 4315: 4313: 4310: 4308: 4305: 4303: 4300: 4298: 4295: 4293: 4290: 4288: 4285: 4283: 4280: 4278: 4275: 4273: 4270: 4269: 4267: 4265: 4259: 4253: 4250: 4248: 4245: 4243: 4240: 4238: 4235: 4233: 4230: 4228: 4225: 4223: 4220: 4218: 4215: 4211: 4208: 4207: 4206: 4203: 4201: 4198: 4196: 4193: 4191: 4188: 4186: 4183: 4181: 4178: 4176: 4173: 4171: 4168: 4166: 4165:Book building 4163: 4161: 4158: 4157: 4155: 4153: 4149: 4146: 4138: 4132: 4129: 4127: 4124: 4122: 4119: 4117: 4114: 4112: 4109: 4107: 4104: 4102: 4099: 4097: 4094: 4092: 4089: 4087: 4084: 4082: 4079: 4077: 4074: 4073: 4071: 4069: 4065: 4061: 4057: 4050: 4045: 4043: 4038: 4036: 4031: 4030: 4027: 4020: 4017: 4014: 4011: 4010: 4002: 3999: 3998: 3989: 3986: 3983: 3980: 3977: 3973: 3969: 3966: 3963: 3960: 3957: 3954: 3950: 3947: 3944: 3942: 3938: 3935: 3933: 3929: 3925: 3921: 3918: 3915: 3912: 3908: 3905: 3903: 3899: 3895: 3892: 3890: 3886: 3883: 3881: 3877: 3873: 3870: 3867: 3863: 3860: 3857: 3856: 3840: 3835: 3831: 3827: 3823: 3818: 3814: 3810: 3806: 3802: 3798: 3794: 3790: 3785: 3781: 3777: 3773: 3769: 3765: 3761: 3757: 3753: 3746: 3741: 3738: 3735: 3728: 3721: 3716: 3712: 3706: 3702: 3697: 3693: 3689: 3685: 3681: 3677: 3673: 3669: 3664: 3661: 3657: 3653: 3648: 3644: 3640: 3636: 3631: 3621:on 2007-12-18 3620: 3616: 3611: 3607: 3602: 3596: 3592: 3587: 3581: 3578: 3577: 3572: 3562:on 2010-06-20 3561: 3557: 3552: 3542:on 2011-07-16 3538: 3531: 3526: 3522: 3517: 3513: 3509: 3505: 3501: 3497: 3493: 3488: 3484: 3480: 3475: 3470: 3465: 3460: 3456: 3451: 3446: 3445:11311/1204834 3441: 3436: 3431: 3427: 3423: 3419: 3415: 3411: 3406: 3405: 3404: 3403: 3402:Applications: 3395: 3389: 3385: 3380: 3376: 3370: 3366: 3365: 3359: 3355: 3349: 3345: 3340: 3336: 3330: 3326: 3321: 3317: 3311: 3307: 3303: 3298: 3294: 3288: 3284: 3283: 3277: 3273: 3267: 3262: 3261: 3254: 3250: 3244: 3240: 3235: 3231: 3225: 3221: 3220: 3214: 3213: 3212: 3211: 3197: 3193: 3192: 3187: 3186: 3181: 3180:Nobel Lecture 3175: 3168: 3167: 3163: 3157: 3151: 3146: 3140: 3136: 3132: 3129: 3123: 3116: 3110: 3103: 3099: 3096:Ronald Fink: 3093: 3080: 3074: 3070: 3066: 3062: 3058: 3051: 3043: 3039: 3034: 3029: 3025: 3021: 3016: 3011: 3007: 3003: 2999: 2992: 2984: 2980: 2976: 2972: 2967: 2966:10044/1/91265 2962: 2958: 2954: 2950: 2946: 2942: 2935: 2927: 2923: 2919: 2915: 2911: 2907: 2903: 2899: 2895: 2888: 2880: 2876: 2872: 2868: 2864: 2860: 2856: 2852: 2848: 2841: 2833: 2829: 2825: 2821: 2817: 2813: 2809: 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