Datar–Mathews method for real option valuation

The DM Method can be extended to an algebraic form creating a generalized form for the DM Option. There are several advantages to the algebraic form, and ancillary simplifications, over the simulated approach as follows:

• Solves the option value explicitly
• Makes transparent the inner workings of option pricing technology
• Applies to many distribution types (lognormal, triangular, uniform, beta)[15]

Fig. 4 Concept of conditional probability distribution and mean of tail

The algebraic form of the DM Option, whether modeled using a lognormal, triangular or other distribution, is fundamentally a simple concept and based on the same computation procedures. It is the conditional expectation of projected future value outcome distribution, ${\displaystyle {\tilde {S}}_{T}}$, less a predetermined purchase cost (strike price or launch cost), ${\displaystyle {\tilde {X}}_{T}}$, multiplied by the probability of that distribution. A conditional expectation is expected value of the truncated distribution (mean of the tail), MT, computed with respect to its conditional probability distribution[16] (Fig. 4).

Fig. 5 Time differentiated discounting appears to shift X relative to S

A financial option valuation (such as Black-Scholes) is based on a lognormal distribution projected from historical asset returns to present time T₀ and using a lognormal distribution modeled by a volatility (finance) factor. A lognormal distribution is a close approximation to the distribution of traded financial asset returns. In contrast, real option valuation is based on a projection of a project’s future values at time T_T, which may or may not be lognormally distributed. For comparison, the following real option example uses a lognormal distribution modeled using a standard deviation, SD, in place of the Black-Scholes volatility factor.

The valuation procedure values the project investment (option purchase, C₀) at T₀. The time differentiated discounting (R and r) results in an apparent shift of ${\displaystyle {\tilde {X}}}$, or the mean ${\displaystyle {\bar {X}}}$, relative to the value outcome distribution, ${\displaystyle {\tilde {S}}}$, or the mean ${\displaystyle {\bar {S}}}$ (Fig. 5).[17][18] This relative shift sets up the conditional expectation of the truncated distribution at T₀. Furthermore, the standard deviation, SD, of the distribution ${\displaystyle {\tilde {S}}}$ is proportionately discounted along with the distribution.

${\displaystyle S_{0}={\bar {S_{T}}}e^{-RT}{\text{, }}SD_{0}=SD_{T}e^{-RT}{\text{ and, }}{X_{0}}={\bar {X_{T}}}e^{-rT}.}$

The standard deviation and mean of the non-logarithmized SD and S values are related to the logarithmic variance and mean, ${\displaystyle \sigma ^{2}{\text{ and }}\mu }$ respectively, as:

${\displaystyle \sigma ^{2}=\ln \left[1+\left({\tfrac {SD_{0}}{S_{0}}}\right)^{2}\right]{\text{, and }}\mu =\ln \left({\tfrac {S_{0}\,^{2}}{\sqrt {SD_{0}\,^{2}+S_{0}\,^{2}}}}\right)=\ln S_{0}-0.5\sigma ^{2}.}$

The conditional expectation of the discounted value outcome is ${\displaystyle MT}$:

${\displaystyle S_{0}\left[{\tfrac {N\left({\tfrac {\sigma ^{2}+\mu -\ln X_{0}}{\sigma }}\right)}{N\left({\tfrac {\mu -\ln X_{0}}{\sigma }}\right)}}\right]{\text{ where }}N(\cdot )}$ is the cumulative distribution function of the standard normal distribution.

The probability of the project being in the money and launched (“exercised”) is ${\displaystyle N\left({\tfrac {\mu -\ln X_{0}}{\sigma }}\right).}$

The project investment (option) value is:

${\displaystyle C_{0}=\left\{S_{0}\left[{\tfrac {N\left({\tfrac {\sigma ^{2}+\mu -\ln X_{0}}{\sigma }}\right)}{N\left({\tfrac {\mu -\ln X_{0}}{\sigma }}\right)}}\right]-X_{0}\right\}N\left({\tfrac {\mu -\ln X_{0}}{\sigma }}\right).}$

Though similar in form to the Black-Scholes option value formula, the DM Option in algebraic form applies variables that are complementary with the structuring of a project valuation. By substituting the standard deviation for the volatility factor (specifically ${\displaystyle \sigma {\sqrt {T}}}$), the two formulas can be shown to be equivalent.

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: ${\displaystyle Z={\tfrac {\left(\ln X_{0}-\mu \right)}{\sigma }}.}$

The conditional expectation of the discounted value outcome is:

${\displaystyle S_{0}\left[{\tfrac {N\left(\sigma -Z\right)}{N\left(-Z\right)}}\right].}$

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

The option value simplifies to:

${\displaystyle C_{0}=\left\{S_{0}\left[{\tfrac {N\left(\sigma -Z\right)}{N\left(-Z\right)}}\right]-X_{0}\right\}N\left(-Z\right)=S_{0}N\left(\sigma -Z\right)-X_{0}N\left(-Z\right).}$

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, ${\displaystyle \textstyle UR=(SD/S)}$, 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 ${\displaystyle \textstyle \left({\tfrac {SD_{T}}{S_{T}}}={\tfrac {SD_{0}}{S_{0}}}\right)}$ with values typically between 0.35 and 1.0 for many multi-year business projects.

Fig. 6 Option value (C₀) vs S₀/X₀ - UR constant

Applying this observation as a constant, K, to the above formulas results in a simpler formulation:

${\displaystyle {\text{Define }}K=\sigma ={\sqrt {\left[\ln(1+UR^{2})\right]}}{\text{, and }}\mu =\ln S_{0}-0.5\mu ^{2}=\ln S_{0}-0.5K^{2}.}$

${\displaystyle Z={\frac {(\ln X_{0}-\mu )}{\sigma }}={\tfrac {\ln \left({\tfrac {X_{0}}{S_{0}}}\right)}{K}}+0.5K{\text{, and }}-Z={\tfrac {\ln \left({\tfrac {S_{0}}{X_{0}}}\right)}{K}}-0.5K.}$

The resulting real option value can be derived simply on a hand-held calculator once K is determined:

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

Assuming the UR is held constant, then the relative value of the option is determined by the ratio of ${\displaystyle S_{0}{\text{ and }}X_{0}}$, which is proportional to the probability of the project being in the money and launched (exercised). Then ${\displaystyle \textstyle C_{0}\approx S_{0}\ x\ \left[0.3\ x\ (S_{0}/X_{0})-k]\right.}$ (Fig. 6) Businesses need apply their own data across similar projects to establish relevant parameter values.

As an additional check on estimating operating profits (${\displaystyle {\tilde {S}}_{0}}$) with uncertainty, an example of ${\displaystyle \textstyle UR=.6}$ results in a value range of:

${\displaystyle {\text{95}}\%{\text{ confidence range }}\approx e^{\left(\mu \mp 2\sigma \right)}\implies {\text{approximate range }}\approx .3\times S_{0}\longleftrightarrow \ \approx 2.5\times S_{0}}$

A value range of approximately an order of magnitude may seem large, even excessive. However, a real option business case is most often applied to multi-year early-stage projects that embody large uncertainties, which therefore is reflected with a wide estimated value range representing worst-to-best case scenarios.

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.[19][20] 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.[21] Regardless of the distribution chosen, the procedure remains the same for a real option valuation.

Fig. 7 Triangular conditional probability distribution

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. 8) 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.[22] Fig. 8 DM Range Option with triangular distribution. (Background: comparable lognormal distribution.) 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 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 95% confidence).[23][24][25][26] This range of estimates results in the eponymous name for the option, the DM Range Option.[27] 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 ${\displaystyle T_{0}}$ discount a, b and m 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 by simulation. Alternatively, without the burden of performing a simulation, applying the average or mean scalar value of the launch cost distribution ${\displaystyle {\bar {X}}_{0}}$ (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 ${\displaystyle MT={\tfrac {\left(2X_{0}+b\right)}{3}}.}$

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. 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:

${\displaystyle P(X_{0}|X_{0}\geq x)={\frac {\left(b-X_{0}\right)^{2}}{\left[\left(b-a\right)\left(b-m\right)\right]}}.}$

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

${\displaystyle C_{0}=\left(MT-X_{0}\right)\cdot P\left(X_{0}\vert X_{0}\geq x\right)=\left(MT-X_{0}\right)\cdot \left\{{\frac {\left(b-X_{0}\right)^{2}}{\left[\left(b-a\right)\left(b-m\right)\right]}}\right\}.}$

The values of the triangular distribution can be estimated to those of the logarithmic distribution by using a range within ${\displaystyle \mp 2}$ standard deviations of its mean (≈95% confidence) as follows:

${\displaystyle a=e^{\mu -2\sigma }{\text{, }}b=e^{\mu +2\sigma }{\text{, and }}m=e^{\mu -\sigma ^{2}}.}$

${\displaystyle {\text{Note (using geometric mean): }}\mu =0.5*\ln(a+b){\text{, }}\sigma =0.25*\ln(b/a){\text{, and }}UR={\sqrt {e^{\sigma ^{2}}-1}}.}$

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.