Real option analysis offers interesting insights on the value of assets and on the profitability of investments, which has made real options a growing field of academic research and practical application. Real option valuation is, however, often found to be difficult to understand and to implement due to the quite complex mathematics involved. Recent advances in modeling and analysis methods have made real option valuation easier to understand and to implement. This paper presents a new method for real option valuation using fuzzy numbers that is based on findings from earlier real option valuation methods and from fuzzy real option valuation. The method is intuitive to understand and far less complicated than any previous real option valuation model to date.Real Options, Fuzzy Numbers, New Method

Collan, Mikael

English

Research Papers in Economics

RESEARCH REPORTS 
 
 
 
 
 
 
 
A New Method for Real Option Valuation using 
Fuzzy Numbers 
 
 
 
Mikael Collan 
 
 
Research Report 7/2008 
 
 
 
 
 
 
 
 
 
Institute for Advanced  
Management Systems Research  
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 RESEARCH REPORTS 
 
 
 
 
 
 
 
A New Method for Real Option Valuation using 
Fuzzy Numbers 
 
 
 
Mikael Collan 
 
 
Research Report 7/2008 
 
 
 
 
 
 
 
 
 
Institute for Advanced  
Management Systems Research  
 
 A New Method for Real Option Valuation using Fuzzy Numbers 
 
Mikael Collan 
 
Institute for Advanced Management Systems Research 
Åbo Akademi University 
Joukahaisenkatu 3-5 A 4
th floor 
20520 Turku, Finland 
mikael.collan@abo.fi 
 
Abstract 
 
Real option analysis offers interesting insights on the value of assets and on the profitability of 
investments, which has made real options a growing field of academic research and practical 
application. Real option valuation is, however, often found to be difficult to understand and to 
implement due to the quite complex mathematics involved. Recent advances in modeling and 
analysis methods have made real option valuation easier to understand and to implement. 
 
This paper presents a new method for real option valuation using fuzzy numbers that is based on 
findings from earlier real option valuation methods and from fuzzy real option valuation. The 
method is intuitive to understand and far less complicated than any previous real option valuation 
model to date. 
 
Keywords: Real Option Valuation, Fuzzy Real Options, Fuzzy Numbers 
 
1. Introduction 
 
Real option valuation (ROV) is based on the observation that the possibilities financial options 
give their holder resemble the possibilities found in real investments, i.e., managerial flexibility, 
e.g., "an irreversible investment opportunity is much like a financial call option" (Pindyck, 
1991). In other words, real option valuation is treating the different types of managerial 
flexibility as options and valuing managerial flexibility with option valuation models. 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 real option value.  
 
Real options are commonly valued with the same methods that have been used to value financial 
options, i.e., with Black-Scholes option pricing formula (Black & Scholes, 1973), with the 
binomial option valuation method (Cox, Ross, & Rubinstein, 1979) , with Monte-Carlo based 
methods (Boyle, 1977), and with a number of later methods based on these. Most of the methods 
are complex and demand a good understanding of the underlying mathematics, issues that make 
their use difficult in practice. Recently, a novel approach to real option valuation was presented 
in (Mathews & Datar, 2007a), (Mathews & Salmon, 2007b), and in (Datar & Mathews, 2004), 
where the real option value is calculated from a pay-off distribution, derived from a probability 
distribution of the NPV for a project that is generated with a (monte-carlo) simulation. The authors show that the results from the method converge to the results from the analytical Black-
Scholes method. The method presented greatly simplifies the calculation of the real option value, 
making it more transparent and brings real option valuation as a method a big leap closer to 
practitioners.  
 
All of the above mentioned models and methods use probability theory in their treatment of 
uncertainty, there are however, other ways than probability to treat uncertainty or imprecise 
future estimates, namely fuzzy logic and fuzzy sets. In classical set theory an element either 
(fully) belongs to a set or does not belong to a set at all. This type of bi-value, or true/false, logic 
is commonly used in financial applications. Bi-value logic, however, presents a problem, because 
financial decisions are generally made under uncertainty. Uncertainty means that it is impossible 
to give absolutely correct precise estimates of, e.g., future cash-flows. Fuzzy sets are sets that 
allow (have) gradation of belonging, such as "a future cash flow at year ten is about x euro". This 
means that fuzzy sets can be used to formalize inaccuracy that exists in human decision making 
and as a representation of vague, uncertain or imprecise knowledge, which human reasoning is 
especially adaptive to. "Fuzzy set-based methodologies blur the traditional line between 
qualitative and quantitative analysis, since the modeling may reflect more the type of information 
that is available rather than researchers' preferences" (Tarrazo, 1997) and indeed in economics 
"The use of fuzzy subsets theory leads to results that could not be obtained by classical 
methods." (Ponsard, 1988). The origins of fuzzy sets date back to an article by Lotfi Zadeh 
(Zadeh, 1965) where he developed an algebra for what he called fuzzy sets. This algebra was 
created to handle imprecise elements in our decision making processes, and is the formal body of 
theory that allows the treatment of practically all decisions in an uncertain environment. 
"Informally, a fuzzy set is a class of objects in which there is no sharp boundary between those 
objects that belong to the class and those that do not" (Bellman & Zadeh, 1970). 
 
DEFINITION. Let          denote a collection of objects (points) denoted generically by x. 
Then a fuzzy set A in X is a set of ordered pairs 
 
Eq. 1.             ,μ       ,        
 
where μ      is termed the grade of membership of x in A, and μ :    is a function from X 
to a space M called the membership space. When M contains only two points, 0 and 1, A is non 
fuzzy and its membership function becomes identical with the characteristic function of a crisp 
set. This means that crisp sets are a subset of fuzzy sets. A fuzzy number is a normal, convex 
fuzzy set whose referential set is the real numbers     .  
 
Fuzzy set theory uses fuzzy numbers to quantify subjective fuzzy observations or estimates. Such 
subjective observations or estimates can be, e.g., estimates of future cash flows from an 
investment. To estimate future cash flows and discount rates "One usually employs educated 
guesses, based on expected values or other statistical techniques" (Buckley, 1987), which is 
consistent with the use of fuzzy numbers. In practical applications the most used fuzzy numbers 
are trapezoidal and triangular fuzzy numbers. They are used, because they make many operations 
possible and are intuitively understandable and interpretable. 
 When we replace non-fuzzy numbers (crisp, single) numbers that are commonly used in financial 
models with fuzzy numbers we can construct models that include the inaccuracy of human 
perception, or ability to forecast, within the (fuzzy) numbers. This makes these models more in 
line with reality, as they do not simplify uncertain distribution-like observations to a single point 
estimate that conveys the sensation of no-uncertainty. Replacing non-fuzzy numbers with fuzzy 
numbers means that the models that are built must also follow the rules of fuzzy arithmetic.  
 
Fuzzy numbers (fuzzy logic) have been adopted to option valuation models in (binomial) pricing 
an option with a fuzzy payoff, e.g., in (Muzzioli & Torricelli, 2000), and in Black-Scholes 
valuation of financial options in, e.g., (Yoshida, 2003). There are also some option valuation 
models that present a combination of probability theory and fuzzy sets, e.g., (Zmeskal, 2001). 
Fuzzy numbers have also been applied to the valuation of real options in, e.g., (Carlsson & 
Fullér, 2003), (Collan, Carlsson, & Majlender, 2003), and  (Carlsson & Majlender, 2005). There 
are also specific fuzzy models for the analysis of the value of optionality for very large industrial 
real investments, e.g.,   (Collan, 2004). 
 
In the following section we will present a new method for valuation of real options from fuzzy 
numbers that is based on the previous literature on real option valuation, especially the findings 
presented in (Mathews et al., 2007a) and on fuzzy real option valuation methods, we continue by 
illustrating the method with a case and close with a discussion and conclusions. 
 
2. New Method for Valuation of Real Options from Fuzzy Numbers 
 
(Mathews et al., 2007b) and (Mathews et al., 2007a) present a practical probability theory based 
method for the calculation of real option value (ROV) and show that the method and results from 
the method are mathematically equivalent to the Black-Sholes formula (Black et al., 1973). The 
method is based on simulation generated probability distributions for the NPV of future project 
outcomes. The method implies that: “the real-option value can be understood simply as the 
average net profit appropriately discounted to Year 0, the date of the initial R&D investment 
decision, contingent on terminating the project if a loss is forecast at the future launch decision 
date.” The project outcome probability distributions are used to generate a payoff distribution, 
where the negative outcomes (subject to terminating the project) are truncated into one chunk 
that will cause a zero payoff, and where the probability weighted average value of the resulting 
payoff distribution is the real option value. 
 
 
We use fuzzy numbers in representing the expected future distribution of possible project costs 
and revenues, and hence also the profitability (NPV) outcomes. When using fuzzy numbers the 
fuzzy NPV itself is a payoff distribution from the project.  
 
The method presented in (Mathews et al., 2007a) implies that the weighted average of the 
positive outcomes of the payoff distribution is the real option value; in the case with fuzzy 
numbers the weighted average is the fuzzy mean value of the positive NPV outcomes (which is 
nothing more than the possibility weighted average). Derivation of the fuzzy mean value is 
presented in (Carlsson & Fullér, 2001). 
  
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
Figure 1. Triangular fuzzy number (a possibility distribution), defined by three points [a, α, β] 
describing the NPV of a prospective project.  
 
This means that calculating the real option value (ROV) from a fuzzy NPV (distribution) is 
straightforward, it is the fuzzy mean of the possibility distribution with values below zero 
counted as zero, i.e., the area weighted average of the fuzzy mean of the positive values of the 
distribution and zero (for negative values).  
 
Real option value is calculated from the fuzzy NPV as following: 
 
Eq. 2.   FROV  = 
           
                                          
 
It is easy to see that when the whole fuzzy number is above zero ROV is the fuzzy mean of the 
number, and when the whole fuzzy number is below zero the ROV is 0.   
 
The components of the new method are simply the observation that real option value is the 
probability weighted average of the positive values of a payoff distribution of a project, which is 
nothing more than the fuzzy NPV of the project, and that for fuzzy numbers the probability 
weighted average of the positive values of the payoff distribution is nothing more than the 
weighted fuzzy mean of the positive values of the fuzzy NPV, when we use fuzzy numbers. 
 
3. Case: Using the New Method in Analyzing Acquisition Synergy as a Real Option 
 
The problem at hand is to evaluate the value of uncertain synergies arising from a corporate 
acquisition that is estimated to last for seven years at maximum. The acquiring company has 
identified three possible scenarios, good, most likely, and bad, for the investments to realize the 
synergies and the synergy benefits. The scenario values are given by managers as non-fuzzy 
numbers, they can have used any type of analysis tools, or models to reach these scenarios. From 
these cost & benefit scenarios three scenarios for the NPV are combined (PV benefits – PV 
1 
0 
α  β  a 
Degree of  
 membership 
0
80% of the area NPV positive outcomes; 
value according to expectation (mean 
value of the positive area; M+) 
Area 
determines 
weight 
20% of the area NPV negative
outcomes; all valued at 0 
M+ investment costs), where the cost cash-flows (CF) are discounted at the risk free rate and the 
benefit CF discount rate is selected according to the risk (risk adjusted discount rate). The NPV 
is calculated for each of the three scenarios separately, i.e., good scenario costs are deducted 
from good scenario benefits; this is not strictly in line with fuzzy arithmetic (as there the largest 
cost would be deducted from the smallest benefit to get the lower bound of the set). 
 
Figure 2. Three NPV scenarios for the duration of the synergies that are used to generate 
(triangular) fuzzy NPV  
 
The resulting fuzzy NPV is the payoff distribution for the synergies investment. The real option 
value for the investment can be calculated from the fuzzy NPV according to the formula 
presented in Eq. 2. In this case, as the whole distribution is above zero the ROV is nothing else 
than the fuzzy mean value of the fuzzy NPV. 
 
The company managers are accustomed to giving information in the form of scenarios (usually 
three) and they have a set of methods for building the scenarios – usually coming from past 
experience and based on looking at issues like the most contributing single issues (or variables) 
and the markets. The scenario approach can be fully omitted and the future forecast can be done 
from the beginning with fuzzy numbers, the end result will be a fuzzy NPV in both cases. 
 
4. Discussion and Conclusions 
 
There is reason to expect that the simplicity of the presented method is an advantage over more 
complex methods. Using triangular and trapezoidal fuzzy numbers make very easy 
implementations possible with the most commonly used spreadsheet software; this opens 
avenues for real option valuation to find its way to more practitioners. The method is flexible as 
it can be used when the fuzzy NPV is generated from scenarios or as fuzzy numbers from the 
beginning of the analysis. Fuzzy NPV is a distribution of the possible values that can take place 
for NPV; this means that it is by definition perceived as impossible at the time of the assessment 
that values outside of the number can happen – this is in line with the situation that real option 
value is zero when all the values of the fuzzy NPV are lower than zero. If we compare this to the 
presented case, we can see that in practice it is often that managers are not interested to use the 
full distribution of possible outcomes, but rather want to limit their assessment to the most 
10,81 24,22 51,36
ROV = 30,98 possible alternatives (and leaving out the “tails” of the distribution). We think that the tails 
should be included in the real option analysis, because even remote possibilities should be taken 
into consideration.  
 
The method brings forth an issue that has not gotten very much attention in academia, the 
dynamic nature of the assessment of investment profitability, i.e., the assessment changes when 
information changes. As cash flows taking place in the future come closer, information changes, 
and uncertainty is reduced this should be reflected in the fuzzy NPV, the more there is 
uncertainty the wider the distribution should be, and when uncertainty is reduced the width of the 
distribution should decrease. Only under full certainty should the distribution be represented by a 
single number, as the method uses fuzzy NPV there is a possibility to have the size of the 
distribution decrease with a lesser degree of uncertainty, this is an advantage vis-à-vis probability 
based methods.  
 
The common decision rules for ROV analysis are applicable with the ROV derived with the 
presented method. We suggest that the single number NPV needed for comparison purposes is 
derived from the (same) fuzzy NPV by calculating the fuzzy mean value. This means that in 
cases when all the values of the fuzzy NPV are greater than zero the single number NPV equals 
ROV, which indicates immediate investment. 
 
We feel that the presented new method opens possibilities for making simpler generic and 
modular real option valuation tools that will help construct real options analyses for systems of 
real options that are present in many types of investments. 
 
5. References 
 
Bellman, R. & Zadeh, L. A. 1970. Decision-making in a fuzzy environment. Management 
Science, 17(4): 141-164. 
Black, F. & Scholes, M. 1973. The Pricing of Options and Corporate Liabilities. Journal of 
Political Economy, 81(May-June ): 637-659. 
Boyle, P. 1977. Options: A Monte Carlo Approach. Journal of Financial Economics, 4(4): 323-
338. 
Buckley, J. 1987. The fuzzy mathematics of finance. Fuzzy Sets and Systems, 21: 257-273. 
Carlsson, C. & Fullér, R. 2001. On possibilistic mean value and variance of fuzzy numbers. 
Fuzzy Sets and Systems, 122: 315-326. 
Carlsson, C. & Fullér, R. 2003. A Fuzzy Approach to Real Option Valuation. Fuzzy Sets and 
Systems, 139: 297-312. 
Carlsson, C. & Majlender, P. 2005. On Fuzzy Real Option Valuation Paper presented at the 9th 
Annual International Conference on Real Options, Paris, France. 
Collan, M., Carlsson, C., & Majlender, P. 2003. Fuzzy Black and Scholes Real Option Pricing. 
Journal of Decision Systems, 12(3-4): 391-416. 
Collan, M. 2004. Giga-Investments: Modelling the Valuation of Very Large Industrial Real 
Investments. Unpublished Dissertation/Thesis, Turku Centre for COmputer Science, Turku. 
Cox, J., Ross, S., & Rubinstein, M. 1979. Option Pricing: A Simplified Approach. Journal of 
Financial Economics, 7: 229-263. Datar, V. & Mathews, S. 2004. European Real Options: An Intuitive Algorithm for the Black-
Scholes Formula. 
Mathews, S. & Datar, V. 2007a. A Practical Method for Valuing Real Options: The Boeing 
Approach. Journal of Applied Corporate Finance, 19(2): 95-104. 
Mathews, S. & Salmon, J. 2007b. Business Engineering: A Practical Approach to Valuing High-
Risk, High-Return Projects Using Real Options. In P. Gray (Ed.), Tutorials in Operations 
Research: Informs. 
Muzzioli, S. & Torricelli, C. 2000. A model for pricing an option with a fuzzy payoff. Fuzzy 
Economics Review, 6(1). 
Pindyck, R. 1991. Irreversibility, Uncertainty, and Investment. Journal of Economic Literature, 
29: 1110-1148. 
Ponsard, C. 1988. Fuzzy mathematical models in economics. Fuzzy Sets and Systems, 28: 273-
283. 
Tarrazo, M. 1997. A Methodology and Model for Qualitative Business Planning. International 
Journal of Business Research, 3(1): 41-62. 
Yoshida, Y. 2003. The valuation of European options in uncertain environment. European 
Journal of Operational Research, 145(1): 221-229. 
Zadeh, L. A. 1965. Fuzzy Sets. Information and Control, 8: 338-353. 
Zmeskal, Z. 2001. Application of the fuzzy-stochastic methodology to appraising the firm value 
as a European call option. European Journal of Operational Research, 135(2): 303-310. 
 
  
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
  
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
  
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
Institute for Advanced Management Systems Research 
Åbo Akademi University 
ICT House 
20520 Åbo, Finland 
 
+358-2-215 4096 (phone) 
+358-2-215 4809 (fax) 
http://iamsr.abo.fi 
 
ISSN 1235-9505 
ISBN 978-952-12-2142-2 
Åbo Akademis tryckeri 
Åbo 2008 

A Fuzzy Approach to Real Option Valuation. Fuzzy Sets and Systems,

A Methodology and Model for Qualitative Business Planning.

A model for pricing an option with a fuzzy payoff.

A Practical Method for Valuing Real Options: The Boeing Approach.

Application of the fuzzy-stochastic methodology to appraising the firm value as a European call option.

Business Engineering: A Practical Approach to Valuing HighRisk, High-Return Projects Using Real Options. In

Decision-making in a fuzzy environment.

Fuzzy Black and Scholes Real Option Pricing.

Fuzzy mathematical models in economics.

Fuzzy Sets.

Giga-Investments: Modelling the Valuation of Very Large Industrial Real Investments. Unpublished Dissertation/Thesis, Turku Centre for COmputer Science,

New Method for Real Option Valuation Using Fuzzy Numbers

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