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. Analysis of these historical trends results in a calculation termed the volatility (finance) factor. For Black-Scholes (BS) the volatility factor is . 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; 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: 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. where MT is the mean of the tail at time T0. Data patterns 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 , less a discounted cost , , multiplied