Datar–Mathews method for real option valuation
The mathematical equation for the DM Method is shown below. The method captures the real option value by discounting the distribution of operating profits at "R", the market risk rate, and discounting the distribution of the discretionary investment at "r", risk-free rate, "before" the expected payoff is calculated. The option value is then the expected value of the maximum of the difference between the two discounted distributions or zero. Fig. 1. formula_1

The differential discount rate for "R" and "r" implicitly allows the DM Method to account for the underlying risk. If "R" > "r", then the option will be risk-averse, typical for both financial and real options. If "R" < "r", then the option will be risk-seeking. If "R" = "r", then this is termed a risk-neutral option, and has parallels with NPV-type analyses with decision-making, such as decision trees. The DM Method gives the same results as the Black–Scholes and the binomial lattice option models, provided the same inputs and the discount methods are used. This non-traded real option value therefore is dependent on the risk perception of the evaluator toward a market asset relative to a privately held investment asset. The DM Method is advantageous for use in real option applications because unlike some other option models it does not require a value for "sigma" (a measure of uncertainty) or for "S0" (the value of the project today), both of which are difficult to derive for new product development projects; see further under real options valuation. Finally, the DM Method uses real-world values of any distribution type, avoiding the requirement for conversion to risk-neutral values and the restriction of a lognormal distribution; see further under Monte Carlo methods for option pricing. Extensions of the method for other real option valuations have been developed such as contract guarantee (put option), Multi-Stage (compound option), Early Launch (American option), and others. The DM Method may be implemented using Monte-Carlo simulation, or in a simplified algebraic or other form (see the Range Option below). Using simulation, for each sample, the engine draws a random variable from both formula_11 calculates their present values, and takes the difference. Fig. 2A. The difference value is compared to zero, the maximum of the two is determined, and the resulting value recorded by the simulation engine. Here, reflecting the optionality inherent in the project, a forecast of a net negative value outcome corresponds to an abandoned project, and has a zero value. Fig. 2B. The resulting values create a payoff distribution representing the economically rational set of plausible, discounted value forecasts of the project at time "T0". When sufficient payoff values have been recorded, typically a few hundred, then the mean, or expected value, of the payoff distribution is calculated. Fig. 2C. The option value is the expected value, the first moment of all positive NPVs and zeros, of the payoff distribution. A simple interpretation is:
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where "operating profit" and "launch costs" are the appropriately discounted range of cash flows to time "T0". The option value can also be understood as a distribution (formula_13) reflecting the uncertainty of the underlying variables. The DM real option can be considered a generalized form for option valuation. Its simulation produces a truncated present value distribution of which the mean value is interpreted to be the option value. With certain boundary conditions, the DM option can be reformulated algebraically as a conditional expectation of a lognormal distribution similar to the form and characteristics of a typical financial option, such as the European, single stage Black-Scholes financial option. This section illustrates the transformation of the DM real option to its algebraic lognormal form and its relationship to the Black-Scholes financial option formula. The process illuminates some of the more technical elements of the option formulation thereby providing further insight to the underlying concepts. The lognormal form of the DM Method remains a simple concept based on the same computation procedures as the simulation form. It is the conditional expectation of the discounted projected future value outcome distribution, formula_2, less a predetermined purchase cost (strike price or launch cost), formula_15, (modeled in this example as a scalar value) multiplied by the probability of that truncated distribution greater than a threshold—nominally 0. A conditional expectation is the expected value of the truncated distribution (mean of the tail), "MT", computed with respect to its conditional probability distribution (Fig. 3). The option calculation procedure values the project investment (option purchase), "C0", at "T0". For the DM option the time differentiated discounting ("R" and "r") results in an apparent shift of the projected value outcome distribution, formula_16, relative to the formula_17, or the scalar mean formula_18 in the example shown in Fig. 4. This relative shift sets up the conditional expectation of the truncated distribution at "T0". In a lognormal distribution for a project future value outcome, formula_2, both the mean, formula_20, and standard deviation, formula_21, must be specified. The standard deviation, formula_21, of the distribution formula_2 is proportionately discounted along with the distribution, formula_24 
The parameters of formula_25, of a lognormal distribution at "T"0 can be derived from the values formula_26 respectively, as:
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The conditional expectation of the discounted value outcome is the mean of the tail "MT":
formula_29 is the cumulative distribution function of the standard normal distribution (N(0,1)). The probability of the project being in the money and launched (“exercised”) is formula_30
The project investment (option) value is:
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The involved lognormal mathematics can be burdensome and opaque for some business practices within a corporation. However, several simplifications can ease that burden and provide clarity without sacrificing the soundness of the option calculation. One simplification is the employment of the standard normal distribution, also known as the Z-distribution, which has a mean of 0 and a standard deviation of 1. It is common practice to convert a normal distribution to a standard normal and then use the standard normal table to find the value of probabilities. Define as the standard normal variable: formula_32
The conditional expectation of the discounted value outcome is: 
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Then probability of the project being in the money and launched (“exercised”) is: formula_34
The Datar-Mathews lognormal option value simplifies to:
formula_35 The Black–Scholes option formula (as well as the binomial lattice) is a special case of the simulated DM real option. With subtle, but notable differences, the logarithmic form of the DM Option can be algebraically transformed into The Black-Scholes option formula. The real option valuation is based on an approximation of the future value outcome distribution, which may be lognormal, at time "TT" projected (discounted) to "T0". In contrast, the Black-Scholes is based on a lognormal distribution projected from historical asset returns to present time "T0". Analysis of these historical trends results in a calculation termed the volatility (finance) factor. For Black-Scholes (BS) the volatility factor is formula_36. The following lognormal distribution with a standard deviation formula_37 is replaced by the volatility factor formula_36.
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The Black-Scholes option value simplifies to its familiar form:
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The terms "N"("d1") and "N"("d2") are applied in the calculation of the Black–Scholes formula, and are expressions related to operations on lognormal distributions; see section "Interpretation" under Black–Scholes. Referring to Fig. 5 and using the lognormal form of the DM Option, it is possible to derive certain insights to the internal operation of an option:
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"N"("-Z") or "N"("d2") is a measure of the area of the tail of the distribution, "MT" (delineated by "X0"), relative to that of the entire distribution, e.g. the probability of tail of the distribution, at time "T0". Fig. 5, Right. The true probability of expiring in-the-money in the real (“physical”) world is calculated at time "T0", the launch or strike date, measured by area of the tail of the distribution. "N"("σ-Z") or "N"("d1") is the value of the option payoff relative to that of the asset. formula_44 where "MT" is the mean of the tail at time "T0". A simplified DM Method computation conforms to the same essential features—it is the conditional expectation of the discounted projected future value outcome distribution, or formula_45, less a discounted cost , formula_46, multiplied by the probability of exercise, formula_47 The value of the DM Method option can be understood as formula_48 This simplified formulation has strong parallels to an expected value calculation. Businesses that collect historical data may be able to leverage the similarity of assumptions across related projects facilitating the calculation of option values. One resulting simplification is the "Uncertainty Ratio", formula_49, which can often be modeled as a constant for similar projects. "UR" is the degree of certainty by which the projected future cash flows can be estimated. "UR" is invariant of time formula_50 with values typically between 0.35 and 1.0 for many multi-year business projects. Applying this observation as a constant, "K", to the above formulas results in a simpler formulation:
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formula_53 is normally distributed and the values can be accessed in a table of standard normal variables. The resulting real option value can be derived simply on a hand-held calculator once K is determined:
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Assuming the "UR" is held constant, then the relative value of the option can simply be determined by the ratio of formula_55, which is proportional to the probability of the project being in the money and launched (exercised). Then formula_56 (Fig. 6) A real option valuation is typically applied when the probability of exercise is approximately less than 50% formula_57 Businesses need apply their own data across similar projects to establish relevant parameter values. Given the difficulty in estimating the lognormal distribution mean and standard deviation of future returns, other distributions instead are more often applied for real options used in business decision making. The sampled distributions may take any form, although the triangular distribution is often used, as is typical for low data situations, followed by a uniform distribution (continuous) or a beta distribution. This approach is useful for early-stage estimates of project option value when there has not been sufficient time or resources to gather the necessary quantitative information required for a complete cash flow simulation, or in a portfolio of projects when simulation of all the projects is too computationally demanding. Regardless of the distribution chosen, the procedure remains the same for a real option valuation. For a triangular distribution, sometimes referred to as three-point estimation, the mode value corresponds to the “most-likely” scenario, and the other two other scenarios, “pessimistic” and “optimistic”, represent plausible deviations from the most-likely scenario (often modeled as approximating a two-sided 1-out-of-10 likelihood or 90% confidence). This range of estimates results in the eponymous name for the option, the DM Range Option. The DM Range Option method is similar to the fuzzy method for real options. The following example (Fig. 7) uses a range of future estimated operating profits of "a" (pessimistic), "b" (optimistic) and "m" (mode or most-likely). For formula_58 discount "a", "b" and "m" by formula_59
The classic DM Method presumes that the strike price is represented by a random variable (distribution formula_60) with the option solution derived by simulation. Alternatively, without the burden of performing a simulation, applying the average or mean scalar value of the launch cost distribution formula_61 (strike price) results in a conservative estimate of DM Range Option value. If the launch cost is predetermined as a scalar value, then the DM Range Option value calculation is exact. The expected value of the truncated triangular distribution (mean of the right tail), is formula_62
The probability of the project being in the money and launched is the proportional area of the truncated distribution relative to the complete triangular distribution. This partial expectation is computed by the cumulative distribution function (CDF) given the probability distribution will be found at a value greater than or equal to "X":
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The DM Range Option value, or project investment, is:
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Use of a DM Range Option facilitates the application of real option valuation to future project investments. The DM Range Option provides an estimate of valuation that differs marginally with that of the DM Option algebraic lognormal distribution form. However, the projected future value outcome, "S", of a project is rarely based on a lognormal distribution derived from historical asset returns, as is a financial option. Rather, the future value outcome, "S", (as well as the strike price, "X", and the standard deviation, "SD"), is more than likely a three-point estimation based on engineering and marketing parameters. Therefore, the ease of application of the DM Range Option is often justified by its expediency and is sufficient to estimate the conditional value of a future project. The fuzzy pay-off method for real option valuation, created in 2009, provides another accessible approach to real option valuation. Though each use differing mathematical methods (Fuzzy: fuzzy logic; DM: numerical simulation and geometry) the underlying principal is strikingly similar: the likelihood of a positive payoff. Separately examining the two factors (possibility/probability, and positive payoff) demonstrates this similarity. The possibility function for the fuzzy pay-off is formula_66. A simple interpretation is the proportionality ratio of the positive area of the fuzzy NPV over the total area of the fuzzy NPV. The probability of the project payoff for the DM Range Option is proportional to the area (CDF) of the positive distribution relative to the complete distribution. This is computed as formula_67 In each the ratios of the areas compute to the same possibility/probability value. The positive payoff of fuzzy pay-off simply is the mean of the positive area of the fuzzy NPV, or formula_68. Likewise, the positive payoff for the DM Range Option is the mean of the right tail ("MT"), or formula_69 less the strike price formula_70 This insight to the mechanics of the two methods illustrates not only their similarity but also their equivalency. In a 2016 article in the "Advances in Decision Sciences" journal, researchers from the Lappeenranta University of Technology School of Business and Management compared the DM Method to the fuzzy pay-off method for real option valuation and noted that while the valuation results were similar, the fuzzy pay-off one was more robust in some conditions. In some comparative cases, the Datar-Mathews Method has a significant advantage in that it is easier to operate and connects NPV valuation and scenario analysis with Monte Carlo simulation (or geometry) technique thus greatly improving intuition in the usage of real options methods in managerial decision and explanation to third parties. Through its simulation interface, the Datar-Mathews Method easily accommodates multiple and sometimes correlated cash flow scenarios, including dynamic programming, typical of complex projects, such as aerospace, that are difficult to model using fuzzy sets. Real options are about objectively valuing innovation opportunities. Vexingly, these opportunities, evanescent and seemingly risky, are often comprehended subjectively. However, both the objective valuation mechanism and the subjective interpretation of results are often misunderstood leading to investment reluctance and potentially undervaluing opportunities. The DM real option employs the objective valuation formula formula_71 where formula_72 is the default threshold when it is economically rational to terminate (abandon) an opportunity event. If a simulation event (‘draw’) calculates a negative outcome (i.e., formula_73 operating profits less than launch costs), then that event outcome should be rationally cut, or terminated, recording a formula_72 residual. Only net positive economic outcomes formula_75 are tallied. This operation leaves the misperception of ‘the odds being stacked’ favoring only positive outcomes seemingly resulting in an abnormally high valuation. However, the formula_76 of the formula formula_77 mathematically calculates the correct option value by adjusting these positive outcomes according to their likelihood, i.e., probability of a success (POS). The actual DM formula is formula_78 where the threshold formula_79 (‘floor’) can assume any value (or alternative formula) including the default formula_72. Using a threshold other than formula_72 transforms the formula into a hurdle-weighted option variation. The result is no longer equivalent to the value of a financial option. Much of the perceived "high" value of a real option valuation is disproportionately located in the far-right end of the tail of the simulation distribution, an area of low probability but high value outcomes. The option valuation reflects the potential opportunity value if the various outcome assumptions are validated. Targeted, incremental investments can validate these low probability assumptions. If not, replace the assumptions with proven ‘plausible’ elements, then recalculate the value based on new learnings. The subjective undervaluation of real options partially can be explained by behavioral sciences. An innovation investor may perceive the initial investments to be potentially at a loss, particularly if the POS is low. Kahneman and Tversky’s Prospect theory proclaims that losses are perceived to have an impact more than twice that of gains for the same value. The result is the loss averse investor will subjectively undervalue the opportunity, and therefore the investment, despite the objective and financially accurate real option valuation. Regret aversion, another behavioral science observation, occurs when an unfounded decision is made to avoid regretting a future outcome. For example, a regret-averse investor decides to invest in a relatively ‘sure bet’ but smaller payoff opportunity relative to an alternative with a significantly higher but presumably uncertain payoff. The regret aversion phenomenon is closely aligned with uncertainty aversion (certainty bias), where the unknown aspects of the innovation opportunity (i.e., newness, lack of control) are rationalized as a hurdle to further investments. The consequences of loss- and regret-averse decision-making are parsimonious investments and underfunding (‘undervaluing’) of promising early-stage innovation opportunities. A savvy investor can overcome the perceived mis-valuation of an option price. Loss aversion registers significantly high when the entire option value is interpreted as investment risk. This emotional response fails to consider that the initial early-stage investments are only a fraction of the entire option value, necessarily targeted to validate the most salient assumption. Similarly regret aversion should not be misconceived as risk aversion because the exposure of small early-stage investments is usually not material. Instead, these initial investments carefully probe the opportunity’s core value while providing a sense of control over an otherwise uncertain outcome. Regret is minimized by the realization that the opportunity development can be terminated if the assumption outcomes are not promising. The investment funds expended are prudently applied only to investigate a promising opportunity, and, in return, are enhanced by the acquired knowledge and the ability to make a better decision. Since individuals are prone to cognitive biases, various intervention strategies are designed to reduce them including expert review along with bias and naïve realism awareness. A phenomenon termed “bias blind spot” succinctly describes an individual’s unconscious susceptibility to biases. This fundamental attribution error remains subconsciously hidden by an illusion of self-introspection, i.e., that we believe, falsely, we have access to our inner intentions or motivations. Biases may be post hoc rationalized away, but nonetheless impact decision-making. To counteract biases, it is insufficient simply be aware of their characteristics, but necessary also to become educated of one’s own introspection illusion.
Comparison to other methods
The Datar–Mathews Method (DM Method) is a method for real options valuation.