Images and Barriers on the road to Real Options Valuation

Traditional methods for evaluating investment decisions, such as Net Present Value, don't properly account for the flexibility inherent in many investment projects. This has been recognized and the attempt to value the such flexibilities is known as Real Options Analysis. This type of investment analysis involves applications of exotic option pricing theory to the evaluation of investment decisions by firms. Many investment projects involve particular types of flexibility and these situations have been identified and recognized as various types of options. This relates to investment decisions about non tradeable assets such as real estate development and mining projects. The decision to delay commencement of a mining project contingent on commodity prices rising enough to make a mining operation viable can be thought of as a type of call option. The decision to temporarily shut down a mining operation due to low commodity prices can be thought of and valued as a type of put option. Some of these options may have barrier features, where the investment project gets cancelled due to commodity prices falling below a critical level. This may the thought of as a "down and out barrier option". Many multi-stage investment decisions can be thought of as compound options, which are options over options. In this paper we consider how to combine the theory of exotic multi-period, multi-asset options and the theory of barrier option pricing to the evaluation of investment decisions. In particular we consider the valuation of compound options with various levels of complexity, and the valuation of barrier versions of these options. We show how to mathematically model such situations, and we derive closed form valuation formulae for evaluating some of them and discuss how to apply numerical methods in other cases. We illustrate the ideas and methods in the context of a hypothetical gold mining project. We derive an analytic formula for the option to delay a mining project which, once commenced, includes the right to further expand production and the right to close down production. The right to delay the commencement of this project is a compound call option over the underlying asset plus a call option plus a put option. The valuation formula for this investment opportunity involves the bivariate normal distribution.