TheDatar–Mathews Method^[1] (DM Method)^[2] is a method forreal options valuation. The method provides an easy way to determine the real option value of a project simply by using the average of positive outcomes for the project. The method can be understood as an extension of thenet present value (NPV) multi-scenarioMonte Carlo model with an adjustment forrisk aversion and economic decision-making. The method uses information that arises naturally in a standarddiscounted cash flow (DCF), orNPV, project financial valuation. It was created in 2000 by Vinay Datar, professor atSeattle University; and Scott H. Mathews,Technical Fellow atThe Boeing Company.

The mathematical equation for the DM Method is shown below. The method captures the real option value by discounting thedistribution ofoperating profits atR, the market risk rate, and discounting the distribution of the discretionary investment atr, 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.^[3][4] Fig. 1.

${\displaystyle C_0=E\left[max\left({\tilde{S}}_T{e}^{-Rt}-{\tilde{X}}_T{e}^{-rt},0\right)\right]}$

• ${\displaystyle {\tilde{S}}_T}$ is arandom variable representing the future benefits, or operating profits at timeT. Thepresent valuation of${\displaystyle {\tilde{S}}_T}$ usesR, a discount rate consistent with the risk level of${\displaystyle {\tilde{S}}_T,}$${\displaystyle {\tilde{S}}_0={\tilde{S}}_T{e}^{-RT}.}$R is therequired rate of return for participation in the target market, sometimes termed thehurdle rate.
• ${\displaystyle {\tilde{X}}_T}$ is a random variable representing thestrike price. The present valuation of${\displaystyle {\tilde{X}}_T}$ usesr, the rate consistent with the risk of capital investment of${\displaystyle {\tilde{X}}_T,}$${\displaystyle {\tilde{X}}_0={\tilde{X}}_T{e}^{-rT}.}$ In many generalized option applications, the risk-free discount rate is used. However other discount rates can be considered, such as the corporate bond rate, particularly when the application is an internal corporate product development project.
• ${\displaystyle C_0}$ is the real option value for a single stage project. The option value can be understood as the expected value of the difference of two present value distributions with an economically rational threshold limiting losses on a risk-adjusted basis. This value may also be expressed as a stochastic distribution.

The differential discount rate forR andr implicitly allows the DM Method to account for the underlying risk.^[5] IfR >r, then the option will berisk-averse, typical for both financial and real options. IfR <r, then the option will be risk-seeking. IfR =r, then this is termed arisk-neutral option, and has parallels with NPV-type analyses with decision-making, such asdecision trees. The DM Method gives the same results as theBlack–Scholes and thebinomial 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 forsigma (a measure of uncertainty) or forS₀ (the value of the project today), both of which are difficult to derive for new product development projects; seefurther underreal options valuation. Finally, the DM Method uses real-world values ofany distribution type, avoiding the requirement for conversion to risk-neutral values and the restriction of alognormal distribution;^[6] seefurther underMonte 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, Early Launch (American option), and others.

The DM Method may be implemented usingMonte-Carlo simulation,^[7] 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${\displaystyle {\tilde{S}}_T\text{ and }{\tilde{X}}_T,}$ calculates their present values, and takes the difference.^[8][9] 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 timeT₀.

Fig. 2A Net profit present-value distribution

Fig. 2B Rational decision distribution

Fig. 2C Payoff distribution and option value

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.^[10]

A simple interpretation is:

${\displaystyle \text{Real option value}=\text{average}\left[max\left(\text{operating profit}-\text{launch costs}\right),0)\right]}$

whereoperating profit andlaunch costs are the appropriately discounted range of cash flows to timeT₀.^[11]

The DM Option can also be interpreted as a logical function. Frequently the simulation is provided by aMonte Carlo function embedded in a spreadsheet, such asMicrosoft Excel. The Excel logic function for a DMcall option is:

${\displaystyle \text{Call option value}=\overline{\text{IF}(({\tilde{S}}_0>{\tilde{X}}_0),({\tilde{S}}_0-{\tilde{X}}_0),0)}}$ , where the overline represents an average value function.

The Excel logic function for a DMput option is:

It is interesting to note that for any operating profit distribution with the same launch cost (strike price) can result in both a call option value as well as a put option value. Changing the call and put options to two different strike prices results in acollar strategy.

Without the averaging function, the option value can also be understood as a distribution (${\displaystyle {\tilde{C}}_0}$) 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 theconditional expectation of the discounted projected future value outcome distribution,${\displaystyle {\tilde{S}}_T}$, less a predetermined purchase cost (strike price or launch cost),${\displaystyle {\overline{X}}_T}$, (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 thetruncated distribution (mean of the tail),MT, computed with respect to itsconditional probability distribution^[12] (Fig. 3).

The option calculation procedure values the project investment (option purchase),C₀, atT₀. For the DM option the time differentiated discounting (R andr) results in an apparent shift of the projected value outcome distribution,${\displaystyle \tilde{S}}$, relative to the${\displaystyle \tilde{X}}$, or the scalar mean${\displaystyle \overline{X}}$ in the example shown in Fig. 4.^[13][14] This relative shift sets up the conditional expectation of the truncated distribution atT₀.

In a lognormal distribution for a project future value outcome,${\displaystyle {\tilde{S}}_T}$, both the mean,${\displaystyle {\overline{S}}_T}$, and standard deviation,${\displaystyle S{D}_T}$, must be specified.^[15] The standard deviation,${\displaystyle S{D}_T}$, of the distribution${\displaystyle {\tilde{S}}_T}$ is proportionately discounted along with the distribution,${\displaystyle S{D}_0=S{D}_T{e}^{-RT}.}$^[16]

The parameters of${\displaystyle \sigma \text{ and }\mu }$, of alognormal atT₀ can be derived from the values${\displaystyle S{D}_0\text{ and }{S}_0}$ respectively, as:

${\displaystyle \sigma =\sqrt{\ln \left[1+{\left(\frac{SD}{S}\right)}^2\right]}\text{ where }\frac{SD}{S}=\frac{S{D}_0}{S_0}=\frac{S{D}_T}{S_T}}$

${\displaystyle \mu =\ln \left(\frac{S_0}{\sqrt{1+{\left(\frac{SD}{S}\right)}^2}}\right)=\ln S_0-0.5\ln \left[1+{\left(\frac{SD}{S}\right)}^2\right]=\ln S_0-0.5{\sigma }^2.}$

Theconditional expectation of the discounted value outcome is the mean of the tailMT:

${\displaystyle E\left[\tilde{S}\mid \tilde{S}\ge X\right]=S_0\left[\frac{N\left(\frac{\mu +{\sigma }^2-\ln X_0}{\sigma }\right)}{N\left(\frac{\mu -\ln X_0}{\sigma }\right)}\right]\text{ where }N(\cdot )}$ is thecumulative distribution function of thestandard normal distribution (N(0,1)).

The probability of the project being in the money and launched (“exercised”) is${\displaystyle N\left(\frac{\mu -\ln X_0}{\sigma }\right).}$

The project investment (option) value is:

${\displaystyle C_0=\left\{S_0\left[\frac{N\left(\frac{{\sigma }^2+\mu -\ln X_0}{\sigma }\right)}{N\left(\frac{\mu -\ln X_0}{\sigma }\right)}\right]-X_0\right\}N\left(\frac{\mu -\ln X_0}{\sigma }\right).}$

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 thestandard 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 thestandard normal table to find the value of probabilities.

Define as thestandard normal variable:${\displaystyle Z=\frac{\left(\ln X_0-\mu \right)}{\sigma }.}$

The conditional expectation of the discounted value outcome is:

${\displaystyle S_0\left[\frac{N\left(\frac{\mu +{\sigma }^2-\ln X_0}{\sigma }\right)}{N\left(\frac{\mu -\ln X_0}{\sigma }\right)}\right]=S_0\left[\frac{N\left(\sigma -Z\right)}{N\left(-Z\right)}\right].}$

Then probability of the project being in the money and launched (“exercised”) is:${\displaystyle N\left(\frac{\mu -\ln X_0}{\sigma }\right)=N\left(-Z\right).}$

The Datar-Mathews lognormal option value simplifies to:

${\displaystyle C_{\text{DM}}=\left\{S_0\left[\frac{N\left(\sigma -Z\right)}{N\left(-Z\right)}\right]-X_0\right\}N\left(-Z\right)=S_{0}N\left(\sigma -Z\right)-X_{0}N\left(-Z\right).}$

TheBlack–Scholes option formula (as well as thebinomial 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.^[17] The real option valuation is based on an approximation of the future value outcome distribution, which may be lognormal, at timeT_T projected (discounted) toT₀. In contrast, theBlack-Scholes is based on a lognormal distribution projected from historical asset returns to present timeT₀.^[18] Analysis of these historical trends results in a calculation termed thevolatility (finance) factor. For Black-Scholes (BS) thevolatility factor is${\displaystyle {\sigma }_{BS}\sqrt{T}}$.^[19][20]

The following lognormal distribution with a standard deviation${\displaystyle \sigma }$ is replaced by the volatility factor${\displaystyle {\sigma }_{BS}\sqrt{T}}$.

${\displaystyle \left(\sigma -Z\right)=\sigma -\frac{\left(\ln X_0-\mu \right)}{\sigma }=\ln S_0-0.5\ln \left[1+{\left(\frac{SD}{S}\right)}^2\right]=\frac{\left[\ln \left(\frac{S_0}{X_T}\right)+\left(r+\frac{{\sigma }_{\text{BS}}^2}{2}\right)T\right]}{\left({\sigma }_{\text{BS}}\sqrt{T}\right)}=d_1\text{ where }\ln X_T=\ln X_0-rT}$

${\displaystyle \left(-Z\right)=-\frac{\left(\ln X_0-\mu \right)}{\sigma }=\frac{\left[\left(\ln S_0-0.5{\sigma }^2\right)-\ln X_0\right]}{\sigma }=\frac{\left[\ln \left(\frac{S_0}{X_T}\right)+\left(r-\frac{{\sigma }_{\text{BS}}^2}{2}\right)T\right]}{\left({\sigma }_{\text{BS}}\sqrt{T}\right)}=d_1-{\sigma }_{\text{BS}}\sqrt{T}=d_2}$

The Black-Scholes option value simplifies to its familiar form:

${\displaystyle C_{\text{BS}}=SN\left(d_1\right)-X_T{e}^{-rT}N\left(d_2\right)}$

The termsN(d₁) andN(d₂) are appliedin the calculation of the Black–Scholes formula, and are expressions related to operations on lognormal distributions;^[21] see section"Interpretation" underBlack–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:

${\displaystyle N\left(\sigma -Z\right)=N\left(d_1\right)}$

${\displaystyle N\left(-Z\right)=N\left(d_2\right)}$

N(-Z) orN(d₂) is a measure of the area of thetail of the distribution,MT (delineated byX₀), relative to that of the entire distribution, e.g. the probability of tail of the distribution, at timeT₀. Fig. 5, Right. The true probability of expiring in-the-money in the real (“physical”) world is calculated at timeT₀, the launch or strike date, measured by area of the tail of the distribution.N(σ-Z) orN(d₁) is the value of the option payoff relative to that of the asset.${\displaystyle N(d_1)=\left[MTxN(d_2)\right]/S_0,}$ whereMT is the mean of the tail at timeT₀.

A simplified DM Method computation conforms to the same essential features—it is the discountedconditional expectation of the discounted projected future value outcome distribution, or${\displaystyle MT}$, less a discounted cost,${\displaystyle X_0}$, multiplied by the probability of exercise,${\displaystyle N\left(-Z\right).}$ The value of the DM Method option can be understood as${\displaystyle C_0=\left(MT-X_0\right)xN\left(-Z\right).}$ This simplified formulation has strong parallels to anexpected 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 theUncertainty Ratio,${\displaystyle UR=(SD/S)}$, 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${\displaystyle \left(\frac{S{D}_T}{S_T}=\frac{S{D}_0}{S_0}\right)}$ 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:

${\displaystyle \text{Define }K=\sigma =\sqrt{\left[\ln (1+U{R}^2)\right]}\text{, and }\mu =\ln S_0-0.5{\sigma }^2=\ln S_0-0.5K^2.}$

${\displaystyle Z=\frac{(\ln X_0-\mu )}{\sigma }=\frac{\ln \left(\frac{X_0}{S_0}\right)}{K}+0.5K\text{, and }-Z=\frac{\ln \left(\frac{S_0}{X_0}\right)}{K}-0.5K.}$

${\displaystyle Z}$ is normally distributed and the values can be accessed in a table ofstandard normal variables. The resulting real option value can be derived simply on a hand-held calculator once K is determined:

${\displaystyle C_0=S_{0}N\left(\sigma -Z\right)-X_{0}N\left(-Z\right)=S_{0}N\left(K-Z\right)-X_{0}N\left(-Z\right).}$

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. Thesampleddistributions may take any form, although thetriangular distribution is often used,as is typical for low data situations, followed by auniform distribution (continuous) or abeta distribution.^[22][23] 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.^[24] Regardless of the distribution chosen, the procedure remains the same for a real option valuation.

Innovation Project Quick Contingent Value Estimate

With most likely PV operating cash flow of $8.5M, but a 3-year capital cost of roughly $20M, the NPV of the important innovation project was deeply negative and the manager is considering abandoning it. Corporate sales analytics estimated with 95% certainty (double-sided 1-in-10 chance) the revenues could be as low as $4M or as high as $34M. (Fig. 7) Because of the large potential upside, the manager believes a small initial investment might resolve the project's downside uncertainties and reveal its potential value.[25] Using the DM Range Option as a guide, the manager calculated the expected contingent value of the project upside to be about $25M [≈ (2*$20M + $34M)/3]. Furthermore, there is a probability of one-in-four {25% ≈ ($34M - $20M)2 /[ ($34M - $4M)($34M-$8.5M)]} that the project revenues will be greater than $20M. With these calculations, the manager estimates the innovation project option value is $1.25M [= ($25M-$20M) * 25%]. Using this value, the manager justifies this initial investment (about 6% of the capital cost) into the project, sufficient to resolve some of the key uncertainties. The project can always be abandoned should the intermediate development results not measure up, but the investment losses will be minimized. (Later using corporate historical data patterns, an analyst converted the values from a three-point estimate to a DM Option calculation, and demonstrated that the result would differ by less than 10%.)

For a triangular distribution, sometimes referred to asthree-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 95% confidence).^{[26][27][28][29]} This range of estimates results in the eponymous name for the option, the DM Range Option.^[30] The DM Range Option method is similar to thefuzzy method for real options. The following example (Fig. 6) uses a range of future estimated operating profits ofa (pessimistic),b (optimistic) andm (mode or most-likely).

For${\displaystyle T_0}$ discounta,b andm by${\displaystyle e^{-RT}\text{ and }{X}_0=X_T{e}^{-rT}.}$

The classic DM Method presumes that the strike price is represented by a random variable (distribution${\displaystyle {\tilde{X}}_0}$) with the option solution derived bysimulation. Alternatively, without the burden of performing a simulation, applying the average or mean scalar value of the launch cost distribution${\displaystyle {\overline{X}}_0}$ (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${\displaystyle MT=\frac{\left(2X_0+b\right)}{3}.}$

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. (See Fig. 16) This partial expectation is computed by thecumulative distribution function (CDF) given the probability distribution will be found at a value greater than or equal toX:

${\displaystyle P(X_0|X_0\ge x)=\frac{{\left(b-X_0\right)}^2}{\left[\left(b-a\right)\left(b-m\right)\right]}.}$

The DM Range Option value, or project investment, is:

${\displaystyle C_0=\left(MT-X_0\right)\cdot P\left(X_0|X_0\ge x\right)=\left(MT-X_0\right)\cdot \left\{\frac{{\left(b-X_0\right)}^2}{\left[\left(b-a\right)\left(b-m\right)\right]}\right\}.}$

${\displaystyle \text{Note: }\mu =0.5\ast \ln (a+b)\text{, }\sigma =0.25\ast \ln (b/a)\text{, and }UR=\sqrt{e^{{\sigma }^2}-1}.}$^[31]

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 itsexpediency and is sufficient to estimate the conditional value of a future project.

Timothy Luehrman in anHBR article states: “In financial terms, a business strategy is much more like a series of options than a series of static cash flows or evendecision trees. Executing a strategy almost always involves making a sequence of risky decisions.”^[32] A multi-stage business strategy valuation can be modeled as a sequence of staged contingent investment decisions structured as a series of DM single-stage options.

In valuing a complex strategic opportunity, a multi-stage, orcompound option,^[33] is a more accurate, but more mathematically demanding, approach than simpler calculations usingdecision tree model,influence diagrams, orlattice/binomial model approaches.^[34][35] Each stage is contingent on the execution or abandonment (gain/success or loss/failure) of the subsequent stage accounting for the investment cost of the preceding stages.The literature references several approaches to modeling a multi-stage option.^{[36][37][38][39][40][41][42][43][44][45]} A three-stage option (1 Proof of concept, 2 Prototype Development, 3 Launch/ Production) can be modeled as:^[46]

The valuation then occurs in reverse order conditioned on success or failure at each stage. The nominal value of this three-stage option is the mean of the multiple (typically several thousand k trials) simulated cash flows.

While the valuation of a multi-stage option is of technical interest, the principal focus of a project manager is the maximization of the probability of success and overall project value. The astute selection of project stagemilestones can simultaneously achieve these goals while also providing project management clarity.^[47][48] Milestone set points are determined by specifying the option payoff for the end stagen value distribution, andbackcasting. Prospective milestones, or value thresholds, for each stagei are designated${\displaystyle P_i^{\ast }}$ (pronounced ‘P-star’). Multiple simulated cash flows, projected from${\displaystyle {\tilde{S}}_0}$, create a pattern of option value responses for each stage revealing prospective candidate milestones.^[49] The simulation evaluates the payoff option values${\displaystyle E\{\left[max\left({\tilde{S}}_i{e}^{-R{t}_0}-{\tilde{X}}_i{e}^{-r{t}_0},0\right)-{\tilde{X}}_{i-1}{e}^{-r{t}_0}\right]\ge P_i^{\ast }\}.}$^[50] A simulation of thousands of trials results in a valuation and ranking of a large sets of data pairs for each stage i: stagei option values mapped to candidate${\displaystyle P_i^{\ast }}$ values.

Aparabolic distribution of data point pairs graphs the sorted range of stagei option values against prospective${\displaystyle P_i^{\ast }}$ milestone values. If selected${\displaystyle P_i^{\ast }}$ threshold is set too low, there are excessive failures to exercise,, and numerically the expected option value is reduced. Alternatively, if selected${\displaystyle P_i^{\ast }}$ threshold is set too high, then there are insufficient instances of successful exercises, and numerically the expected option value is reduced again. The optimal milestone${\displaystyle P_i^{\ast \ast }}$ (‘P-double star’) value that emerges during simulation maximizes the overall project option value by balancing gains and losses.

A three-stage option optimized for management by milestone and value maximization can be modeled as:^[51]

^[52]

Or, succinctly,

${\displaystyle C_0=E\left[max\left({\tilde{S}}_n{e}^{-R{t}_0}-{\tilde{X}}_n{e}^{-r{t}_0},0\right)-{\tilde{X}}_{n-i}{e}^{-r{t}_0}\right]\times \mathbb{P}\left({\tilde{S}}_1^{n-1}{e}^{-R{t}_0}\ge P_1^{n-1\ast \ast }\right).}$^[53]

Note that typically${\displaystyle P_i^{\ast \ast }\gg {\tilde{X}}_i{e}^{-r{t}_0}.}$ The insertion of these carefully determined conditional milestones increases the overall value of the nominal multi-stage option because each successive stage has been optimized to maximize the option value. By use of selected milestones, the project manager achieves the goals of increasing the probability of success and overall project value while also reducing the project managerial burden.

Many early-stage projects find that the dominant unknown values are the first-order range estimates of the major components of operating profits: revenue and manufacturingcost of goods sold (COGS). In turn the uncertainty about revenue is driven byguesstimates of either market demand price or size. Market price and size can be estimated independently, though coupling them together in amarket demand relationship is a better approach. COGS, the total of cost of product quantity to be sold, is the final component and trends according to an experience orlearning curve cost relationship linked to market size. The interplay of these three market elements within a DM Real Options simulation, even with early-stage ranges, can reduce uncertainty for project planning by yielding reasonably narrowed target estimates for product pricing and production size that maximizes potential operating profits and improves option value.

A market pricedemand curve graphs the relationship of price to size, or quantity demanded. Thelaw of demand states there is an inverse relationship between price and quantity demanded, or simply as the price decreases product quantity demanded will increase. A second curve, the manufacturing cost graph, models the learning curve effect illustrating the relationship between the quantity of goods produced and the efficiency gains of that production. Fig. 12. Mathematically, thelearning curve takes the form of a power function.^[54]

A demand curve can be realistically modeled using an inverse lognormal distribution which convolves the market price distribution estimate with the market size range.^[55][56] A demand curve deftly models highlydifferentiated markets which through pricing distinguish selective product or service characteristics such as quality or grade, functional features, and availability, along with quantity sold. Examples are automobiles, shoes, smart phones, and computers.Airfare markets are highly differentiated where demand pricing and quantity sold are dependent on seasonality, day of week, time of day, routing, sale promotions, and seating or fare class. The airfare demand distribution pattern is well represented with an inverse lognormal distribution as shown in Fig. 13.

Curves for all the above components, market price, size, and COGS, can be simulated with variability to yield an optimal operating profit input for the real option calculation (Fig. 14). For example, the simulation results represented in Fig. 15. indicate ranges for price and unit quantity that potentially will maximize profitability. Extracted from these first-order range estimates, a selection of the peak (frequency) values identifies a significantly narrowed spread of promising estimates. Knowing these optimal value spreads substantially reduces uncertainty and provides a better, more targeted set of parameters from which to confidently base innovation development plans and option value.

Thefuzzy 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${\displaystyle \frac{A(Pos)}{A(Pos)+A(Neg)}}$. A simple interpretation is theproportionalityratio 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${\displaystyle \frac{{\left(b-X_0\right)}^2}{\left[\left(b-a\right)\left(b-m\right)\right]}.}$ 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${\displaystyle E[A^{+}]}$. Likewise, the positive payoff for the DM Range Option is the mean of the right tail (MT), or${\displaystyle \frac{\left(2X_0+b\right)}{3}}$ less the strike price${\displaystyle X_0}$ This insight to the mechanics of the two methods illustrates not only their similarity but also their equivalency.

In a 2016 article in theAdvances in Decision Sciences journal, researchers from theLappeenranta University of Technology School of Business and Management compared the DM Method to thefuzzy 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.^[57] 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.^[58] 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.^[59]

Real options are about objectively valuinginnovation 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${\displaystyle C_0=E[max(...,0)],}$ where${\displaystyle 0}$ is the default threshold when it is economicallyrational to terminate (abandon) an opportunity event. If a simulation event (‘draw’) calculates a negative outcome (i.e.,${\displaystyle [{\tilde{S}}_T{e}^{-Rt}\le {\tilde{X}}_T{e}^{-rt}],}$ operating profits less than launch costs), then that event outcome should be rationally cut, or terminated, recording a${\displaystyle 0}$ residual. Only net positive economic outcomes${\displaystyle [{\tilde{S}}_T{e}^{-Rt}>{\tilde{X}}_T{e}^{-rt}]}$ 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${\displaystyle E}$ of the formula${\displaystyle C_0=E[max(...,0)]}$ mathematically calculates the correct option value by adjusting these positive outcomes according to their likelihood, i.e., probability of a success (POS).^[60]

The actual DM formula is${\displaystyle C_0=E[max(...,?)],}$ where the threshold${\displaystyle ?}$ (‘floor’) can assume any value (or alternative formula) including the default${\displaystyle 0}$. Using a threshold other than${\displaystyle 0}$ 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 perceivedhigh 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 bybehavioral sciences. An innovation investor may perceive the initial investments to be potentially at a loss, particularly if the POS is low.Kahneman and Tversky'sProspect theory^[61] proclaims that losses are perceived to have an impact more than twice that of gains for the same value.^[62][63] The result is theloss averse investor will subjectively undervalue the opportunity, and therefore the investment, despite the objective and financially accurate real option valuation. The pursuit of Prospect Theory has recently led to the related fields ofbehavioral economics andbehavioral finance.

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 withuncertainty 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 assumptions. Similarly regret aversion should not be misconceived asrisk 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.

Since individuals are prone tocognitive biases, various intervention strategies are designed to reduce them including expert review along with bias andnaïve realism awareness.^[64] A phenomenon termed “bias blind spot” succinctly describes an individual's unconscious susceptibility to biases.^[65] Thisfundamental attribution error remains subconsciously hidden by anillusion 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.^[66]

Many companies now are experimenting with or implementing AI for new products, financial risk management. and innovation applications.^[67] However, managers should be aware of the central contradictions between the programmatic operations ofAIs (orLLMs) and the desire to create something new and different. In short, innovation and real option analysis are focused on event tails—that is, rare, low-probability out-of-the-norm opportunities—and therefore shines a light on a deep distinction between how LLMs are designed to operate and how analysts want to use them for tail analysis or “what if?” reasoning.

Normal LLM operation models are built to provide the most probable (i.e., “mode”) continuation or answer, based on training over vast text corpora. When you ask a question, the model predicts what is most likely to appear next according to patterns in the data. Some systems introducerandomness (temperature, top-k sampling, etc.), letting them sample from plausible “minority” answers—but the odds still heavily favor the head/tail breakdown of their probability distribution. The default LLM approach is to minimize risk of error, bias toward the statistically “safest” (i.e., modal) answer, and avoid “surprising” (low-probability) completions.

A contradiction naturally arises when attempting to use LLMs to analyze event tails or forecast low-probability, high-impact outcomes. The “tails” of the model’s output distribution are the very unlikely predictions—thewords or tokens that almost never come next. When training information is sparse or absent (“the tail” of what models have seen/learned), the model may “improvise” to maintain plausibility or fluency, which can lead to elegant nonsense. In AI terms this is a confabulation or“hallucination”, a creation of false content and an apparently plausible—sometimes even perfectly grammatically or methodically correct—answer, assertion, or citation that is not actually grounded in facts, training data, or logic.

This represents not a computational contradiction, but rather a mismatch and limitation of the use-case for LLMs. For predictive accuracy, the system’s job is to maximize the likelihood of its answers and almost always give “central” (frequentist, mode) answers. But for exploring tails or new innovative concepts, the LLMs scoring of how likely or unlikely something is can model rare outputs (e.g., less likely completions) that are not tied to real-world probabilities unless explicitlyinstructed or grounded.

Human analysts want to understand both central tendencies and tail risks, that is the full distribution, and want tail outputs to be meaningful, not just made-up. For rigorous, trustworthytail analysis, use coded simulations or grounded tools—and prompt your LLM to distinguish between data-driven and speculative reasoning.^[68] There are several best practices to elicit morereliable information on tail distributions.

1. Explicitly ask LLMs to list unlikely, edge, or counterfactual cases (“Give me rare, edge-case, or counterintuitive answers, and explain the reasoning or likelihood.”)

2. Use tool-augmented or code models for empirical tail analysis (using numeric outputs).

3. Apply external methods for uncertainty quantification (e.g., re-ask/ensemble, leverage retrieval-augmented methods, or inspect output distributions and exponentiate the logprobs parameter to get probabilities).

4. Demand reasoning and context to separate plausible edge cases from hallucinations. Encourage referencing history (grounding) and labeling speculation explicitly.

LLMs are like masters of plausible conversation and “average case” reasoning. But if you want a system to be a risk analyst, you need something that rigorously checks, simulates, and grounds its tail scenarios—either a different model, a code tool, or careful prompt engineering. Inquires to AI itself can provide structured guidance to create probing inquiries and rendering code. Turns out AI can memorize and pattern-match, but what it can’t do is reason like the human mind.

• Real options
• Monte Carlo methods in finance
• Financial models

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