The Datar–Mathews Method (DM Method)U.S. Patent No. 6,862,579 (issued March 1, 2005). 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 (NPV) 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 (DCF), 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 capital 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 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, Early Launch (American option), and others.

Implementation

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 calculates their present values, and takes the difference.Business Engineering: A Practical Approach to Valuing High-Risk, High-Return Projects Using Real Options Tutorials in Operations Research 2007, Operations Research Tools and Applications: Glimpses of Future Technologies, p157–175Business Engineering: A Practical Approach to Valuing High-Risk, High-Return Projects Using Real Options INFORMS Annual Meeting, November 4–7, 2007 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.

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

, where the overline represents an average value function.

The Excel logic function for a DM put 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 a collar strategy.

Without the averaging function, the option value can also be understood as a distribution () reflecting the uncertainty of the underlying variables.

DM Option Variations

Algebraic lognormal form

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, , less a predetermined purchase cost (strike price or launch cost), , (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, , relative to the , or the scalar mean 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, , both the mean, , and standard deviation, , must be specified. The standard deviation, , of the distribution is proportionately discounted along with the distribution, Hereafter, variables are assumed to be scaler unless otherwise specified as a distribution indicated by a tilde

The parameters of , of a lognormal at T0 can be derived from the values respectively, as:

The conditional expectation of the discounted value outcome is the mean of the tail MT: 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

The project investment (option) value is:

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 (“exercised”) is:

The Datar-Mathews lognormal option value simplifies to:

Transformation to the Black-Scholes Option

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.The Black-Scholes model classically assumed that a lognormal distribution most closely approximated the statistical distribution of an asset’s returns. This assumption conveniently simplified the mathematics of the option pricing formula and it remains a useful approximation. In reality security prices do not follow a strict stationary log-normal process. The value results using the Black–Scholes model differ slightly from real world financial option prices in part because of simplifying assumptions of the model. Analysis of these historical trends results in a calculation termed the volatility (finance) factor. For Black-Scholes (BS) the volatility factor is .Variance of a lognormal distribution is , and its standard deviation is . Variance of stock prices are assumed to follow a Wiener Process or geometric Brownian motion proportional to time and its standard deviation (or volatility) is . The relationship of Black Scholes (BS) volatility (annualized) and the logarithmic standard deviation is then: Alternatively, where S is the mean and SD is the standard deviation of the present value lognormal distribution.

The following lognormal distribution with a standard deviation is replaced by the volatility factor .

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

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;Don Chance (2011). Derivation and Interpretation of the Black–Scholes Model . 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: