Datar–Mathews method for real option valuation
The Datar–Mathews Method is a method for real 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 the net present value multi-scenario Monte Carlo model with an adjustment for risk aversion and economic decision-making. The method uses information that arises naturally in a standard discounted cash flow, or NPV, project financial valuation. It was created in 2000 by Vinay Datar, professor at Seattle University; and Scott H. Mathews, Technical Fellow at The Boeing Company.
Method
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.- is a random variable representing the future benefits, or operating profits at time T. The present valuation of uses R, a discount rate consistent with the risk level of. R is the required rate of return for participation in the target market, sometimes termed the hurdle rate.
- is a random variable representing the strike price. The present valuation of uses r, the rate consistent with the risk of investment of. 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.
- 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 DM Method is advantageous for use in real option applications because unlike some other option models it does not require a value for sigma or for S0, 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, Multi-Stage, Early Launch, and others.
Implementation
The DM Method may be implemented using Monte-Carlo simulation, or in a simplified, algebraic or other form.Using simulation, for each sample, the engine draws a random variable from both ST and XT, 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:
where operating profit and launch costs are the appropriately discounted range of cash flows to time T0.
Interpretation
Under certain constraints, the framework of a project investment problem structured for the Datar–Mathews Method can be converted to an equivalent framework structured for the Black–Scholes formula. Fig. 3, Left. The Black–Scholes option pricing model is constrained to a lognormal distribution for the asset value, S, typical of traded financial options, and requires a scalar value for both S0 and XT, and sigma, a measure of volatility of the asset S. Assume a project investment problem with a forecasted lognormal asset value distribution with mean and standard deviation σT. The equivalent Black–Scholes values are:The terms N and N 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. The Datar–Mathews Method does not use N or N, but instead typically solves the option problem by means of Monte Carlo simulation applicable to many different types of distributions inherent in real option contexts. When the Datar–Mathews Method is applied to assets with lognormal distributions, it becomes possible to visualize graphically the operation of N and N.
N is a measure of the area of the tail of the distribution relative to that of the entire distribution, e.g. the probability of tail of the distribution, at time T0. The tail of the distribution is delineated by, the present value of the strike price. Fig. 3, Right. The true probability of expiring in-the-money in the real world is calculated at time T, the launch date, measured by area of the tail of the distribution delineated by XT. N is the value of the option payoff relative to that of the asset;, where MT is the mean of the tail at time T0. Using the DM Method, the value of a call option can be understood as.
DM Option
Algebraic form (Lognormal)
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
A financial option valuation is based on a lognormal distribution projected from historical asset returns to present time T0 and using a lognormal distribution modeled by a volatility 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 TT, 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 at T0. The time differentiated discounting results in an apparent shift of, or the mean, relative to the value outcome distribution,, or the mean . This relative shift sets up the conditional expectation of the truncated distribution at T0. Furthermore, the standard deviation, SD, of the distribution is proportionately discounted along with the distribution.
The standard deviation and mean of the non-logarithmized SD and S values are related to the logarithmic variance and mean, respectively, as:
The conditional expectation of the discounted value outcome is :
The probability of the project being in the money and launched is
The project investment value is:
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, the two formulas can be shown to be equivalent.
Data patterns
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:
The conditional expectation of the discounted value outcome is:
Then probability of the project being in the money and launched is:
The option value simplifies to:
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,, 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 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:
The resulting real option value can be derived simply on a hand-held calculator once K is determined:
Assuming the UR is held constant, then the relative value of the option is determined by the ratio of, which is proportional to the probability of the project being in the money and launched. Then Businesses need apply their own data across similar projects to establish relevant parameter values.
As an additional check on estimating operating profits with uncertainty, an example of results in a value range of:
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.
DM Range Option – (Triangular)
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 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.| 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 the revenues could be as low as $4M or as high as $34M. 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. Using the DM Range Option as a guide, the manager calculated the expected contingent value of the project upside to be about $25M . Furthermore, there is a probability of one-in-four that the project revenues will be greater than $20M. With these calculations, the manager estimates the innovation project option value is $1.25M . Using this value, the manager justifies this initial investment 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. |
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. 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 uses a range of future estimated operating profits of a, b and m.
For discount a, b and m by
The classic DM Method presumes that the strike price is represented by a random variable 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 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, is
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 given the probability distribution will be found at a value greater than or equal to X:
The DM Range Option value, or project investment, is:
The values of the triangular distribution can be estimated to those of the logarithmic distribution by using a range within standard deviations of its mean as follows:
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,, 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.