Real options analysis and method of fuzzy logic

Real options are similar to financial options, except that it applies the theory of options to real life projects.

Real options analysis is looking at investment projects and assets as possibilities and analyzing their acquisition, project start, or making changes within projects as exercising an option. Real option valuation (ROV) is valuing these possibilities as options. Real options are useful both, as a mental model for strategic and operational decision- making and as a valuation and numerical analysis tool. This paper concentrates on the use of real options in numerical analysis, and particularly on the derivation of a numerical monetary real option value for a given investment opportunity, or identified managerial flexibility.

Real options are most commonly valued with the same methods that have been used to value financial options since the 1970s, i.e., with the Black-Scholes option pricing formula [Black and Scholes 1973] with the binomial option valuation method [Cox, Ross et al. 1979], and with Monte-Carlo based methods that use the logic of the Black-Scholes formula [Boyle 1977]. These models are based on the assumption that they can quite accurately mimic the underlying markets as a process, an assumption that may hold for some quite efficiently traded financial securities, but may not hold for real investments that do not have existing markets, or that have markets that can by no means be said to exhibit even weak market efficiency. That is, the construct of the “classical” models requires market completeness and efficiency.

Recently, a novel approach to real option valuation was presented by Datar and Mathews [Datar and Mathews 2004; Mathews and Datar 2007; Mathews and Salmon 2007], where the real option value is calculated from a pay-off distribution

(a distribution of the future expected values from a real asset), in their case a probability distribution of the NPV for a project that is generated with a (MonteCarlo) simulation. Datar and Mathews show that the results from the method converge to the results from the analytical Black-Scholes method.

The defining difference between financial options and real options is the nature and liquidity of the underlying asset. Financial assets such as equities, bonds or foreign exchange are traded in a transparent market with a close-to-perfect pricing (depending on the fundamental view on how perfect the market is). These attributes of the underlying assets make it possible to trade the matching options at market prices on exchanges in a similar way. Real options are on the contrary written on very illiquid assets which are often unique and consequently very difficult to price at a market price. It can be real options on underlying assets such as different mineral mines, oil fields or life science development projects. For clarity, the main differences between financial options and real options can be seen below:

Financial Options Real Options

‐ Short Maturity, usually in months.

‐ Underlying variable driving its value is equity price or price of a financial asset.

‐ Cannot control option value by manipulating stock prices.

‐ Values are usually small.

‐ Competitve or market affects are irrelevant to its value and pricing.

‐ Have been around and traded for more than four decades.

‐ Usually solved using closed form partial differential         equations         and simulation/variance reduction techniques for exotic options.

‐ Marketable and traded security with comparables and pricing info.

‐ Management assumptions and actions have no bearing on valuation.

‐ Longer maturity, usually in years.

‐ Underlying variables are free cash flows, which in turn are driven by competition, demand and management.

‐ Can increase strategic option value by management decisions and flexibility.

‐ Major million and billion dollar decisions.

‐ Competition and market drive the value of a strategic option

‐ A recent development in corporate finance within the last decades.

‐ Usually solved using closed‐form equations and binominal lattices with simulation of the underlying variables, not on the option analysis.

‐ Not traded and proprietary in nature, with no market comparables.

‐ Management assumptions and actions drive the value of a real option.

FUZZY SETS AND NUMBERS

The basic concept of fuzzy sets and numbers originates in the classical set theory developed by Zadeh in 1965 [Zadeh, 1965]. The difference from the classical set theory is the value assigning of the membership grade, which previously was assigned using the grade of either one (for complete membership) or zero (for zero membership). Zadeh introduced the concept of not determining the value from a bivalent point of view but instead creating a continuum of grades to describe the grade of membership, hence making it possible to have e.g. a 0.6

grade of membership, defined by

Л(х) = Х -> [ОД]

, where A(x) represents a fuzzy subset A of a non-empty set X in the interval [0, 1].

In other words the notation of A(x) simply represents a function, the membership function of A, while the X represents the universe for the fuzzy subset. The degree to which the statement “x is in A” (where x denotes a fuzzy quantity and A is the membership function) is true, is determined by finding the ordered pair (x, A(x)). This degree of truth is the second element of the ordered pair. All these ordered pairs completely define A, which can be written as

А = {(хМ(х)|х Е X}

A γ-level set (or γ-cut off) of a fuzzy set A of X is the non-fuzzy (crisp) set denoted by (A)γ which separates the possibility distribution in positive outcomes and negative outcomes. As negative outcomes in the real option world are considered having a value of zero due to the managerial flexibility that makes it possible to terminate projects with a projected negative NPV, only the positive area of the possibility distribution is needed to calculate the value of the real option.

So if A is a fuzzy number the following notation is introduced

HiQj = minf^l", «205 = тпях[Лр where ^(у) denotes the left-hand side while a 2(y) denotes to the righthand side of the у - cut off, у e [0,1].

ОД= \ у(яДу)+a//l)d/= ^---- 2------

Ydy

^(^jfrQ2152^-^]' + [ьк2г1^_в1^)Г)<*" U'^W-^3^

These findings will be used to calculate the fuzzy real options value.