A key input parameter of any ROA problem is the volatility factor that represents the uncertainty associated with the underlying asset value. Typically, it is calculated as an aggregate factor built from many of the uncertainties that contribute to it. For example, the aggregate volatility used in the ROA of a product development project is representative of and a function of multiple uncertainties, including the unit price, number of units sold, unit variable cost, etc. If one of the sources of uncertainty has a significant impact on the options value compared to the others or if management decisions are to be tied to a particular source of uncertainty, you may want to keep the uncertainties separate in the options calculations. For instance, if you own a lease on an undeveloped oil reserve, you face two separate uncertainties: the price of oil and the quantity of oil in the reserve. You may want to treat them separately in evaluating the ROV.

When multiple sources of uncertainty are considered, the options are called rainbow options, and this warrants the use of different volatility factors one for each source of uncertainty in the options calculations. The options solution method is basically the same as for a single volatility factor except that it involves a quadrinomial tree instead of a binomial. This is because the asset can take one of four values as you move from one node to the nodes of the next time step in the...