A fuzzy approach to real option valuation ∗

Christer Carlssonchrister.carlsson@abo.fi

Robert Fullerrfuller@abo.fi

Abstract

To have a real option means to have the possibility for a certain period toeither choose for or against making an invetsment decision, without bindingoneself up front. The real option rule is that one should invest today only ifthe net present value is high enough to compensate for giving up the valueof the option to wait. Because the option to invest loses its value when theinvestment is irreversibly made, this loss is an opportunity cost of investing.The main question that a management group must answer for a deferrable in-vestment opportunity is: How long do we postpone the investment, if we canpostpone it, up to T time periods? In this paper we shall introduce a (heuris-tic) real option rule in a fuzzy setting, where the present values of expectedcash flows and expected costs are estimated by trapezoidal fuzzy numbers.We shall determine the optimal exercise time by the help of possibilistic meanvalue and variance of fuzzy numbers.

1 Probabilistic real option valuation

Options are known from the financial world where they represent the right to buyor sell a financial value, mostly a stock, for a predetermined price (the exerciseprice), without having the obligation to do so. The actual selling or buying of theunderlying value for the predetermined price is called exercising your option. Onewould only exercise the option if the underlying value is higher than the exerciseprice in case of a call option (the right to buy) or lower than the exercise prise inthe case of a put option (the right to sell). In 1973 Black and Scholes [4] madea major breakthrough by deriving a differential equation that must be satisfied bythe price of any derivative security dependent on a non-dividend paying stock. Forrisk-neutral investors the Black-Scholes pricing formula for a call option is

C0 = S0N(d1)−Xe−rTN(d2),∗The final version of this paper appeared in: Fuzzy Sets and Systems, 139(2003) 297-312.

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where

d1 =ln(S0/X) + (r + σ2/2)T

σ√T

, d2 = d1 − σ√T ,

and where, C0 is the option price, S0 is the current stock price, N(d) is the prob-ability that a random draw from a standard normal distribution will be less thand, X is the exercise price, r is the annualized continuously compounded rate on asafe asset with the same maturity as the expiration of the option, T is the time tomaturity of the option (in years) and σ denotes the standard deviation of the annu-alized continuously compounded rate of return of the stock. In 1973 Merton [10]extended the Black-Scholes option pricing formula to dividends-paying stocks as

C0 = S0e−δTN(d1)−Xe−rTN(d2) (1)

where,

d1 =ln(S0/X) + (r − δ + σ2/2)T

σ√T

, d2 = d1 − σ√T

where δ denotes the dividends payed out during the life-time of the option.Real options in option thinking are based on the same principles as financial

options. In real options, the options involve ”real” assets as opposed to financialones [1]. To have a ”real option” means to have the possibility for a certain periodto either choose for or against making an invetsment decision, without bindingoneself up front. For example, owning a power plant gives a utility the opportunity,but not the obligation, to produce electricity at some later date.

Real options can be valued using the analogue option theories that have beendeveloped for financial options, which is quite different from traditional discountedcashflow investment approaches. In traditional investment approaches investmentsactivities or projects are often seen as now or never and the main question iswhether to go ahead with an investment yes or no [3].

Formulated in this way it is very hard to make a decision when there is un-certainty about the exact outcome of the investment. To help with these toughdecisions valuation methods as Net Present Value (NPV) or Discounted Cash Flow(DCF) have been developed. And since these methods ignore the value of flexi-bility and discount heavily for external uncertainty involved, many interesting andinnovative activities and projects are cancelled because of the uncertainties.

However, only a few investment projects are now or never. Often it is possibleto delay, modify or split up the project in strategic components which generateimportant learning effects (and therefore reduce uncertainty). And in those casesoption thinking can help [8]. The new rule, derived from option pricing theory(1), is that you should invest today only if the net present value is high enoughto compensate for giving up the value of the option to wait. Because the option

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to invest loses its value when the investment is irreversibly made, this loss is anopportunity cost of investing. Following Leslie and Michaels [7] we will computethe value of a real option by

ROV = S0e−δTN(d1)−Xe−rTN(d2) (2)

where,

d1 =ln(S0/X) + (r − δ + σ2/2)T

σ√T

, d2 = d1 − σ√T

and where ROV denotes the current real option value, S0 is the present value ofexpected cash flows, X is the (nominal) value of fixed costs, σ quantifies the un-certainty of expected cash flows, and δ denotes the value lost over the duration ofthe option.

We illustrate the main difference between the traditional (passive) NPV deci-sion rule and the (active) real option approach by an example quoted from ([7],page 10):

. . . another oil company has the opportunity to acquire a five-year li-cence on block. When developed, the block is expected to yield 50million barrels of oil. The current price of a barell of oil from thisfield is $10 and the present value of the development costs is $600million. Thus the NPV of the project opportunity is

50 million × $10 - $600 million = -$100 million.

Faced with this valuation, the company would obviously pass up theopportunity. But what would option valuation make of the same case?To begin with, such a valuation would recognize the importance of un-certainty, which the NPV analysis effectively assumes away. There aretwo major sources of uncertainty affecting the value of the block: thequantity and the price of the oil. One can make a reasonable estimateof the quantity of the oil by analyzing historical exploration data ingeologically similar areas. Similarly, historical data on the variablityof oil prices is readily available.

Assume for the sake of argument that these two sources of uncertaintyjointly result in a 30 percent standard deviation (σ) around the growthrate of the value of operating cash inflows. Holding the option alsoobliges one to incur the annual fixed costs of keeping the reserve active- let us say, $15 million. This represents a dividend-like payout ofthree percent (i.e. 15/500) of the value of the assets.

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We already know that the duration of the option, T , is five years andthe risk-free rate, r, is 5 percent, leading us to estimate option value at

ROV = 500× e−0.03×5 × 0.58− 600× e−0.05×5 × 0.32= $251 million - $151 million = $100 million.

The main question that a company must answer for a deferrable investmentopportunity is: How long do we postpone the investment up to T time periods? Toanswer this question, Benaroch and Kauffman ([2], page 204) suggested the fol-lowing decision rule for optimal investment strategy: Where the maximum deferraltime is T , make the investment (exercise the option) at time t∗, 0 ≤ t∗ ≤ T , forwhich the option, Ct∗ , is positive and attends its maximum value,

Ct∗ = maxt=0,1,...,T

Ct = Vte−δtN(d1)−Xe−rtN(d2), (3)

where

Vt = PV(cf0, . . . , cfT , βP )− PV(cf0, . . . , cft, βP ) = PV(cft+1, . . . , cfT , βP ),

that is,

Vt = cf0 +T∑j=1

cfj(1 + βP )j

− cf0 −t∑

j=1

cfj(1 + βP )j

=T∑

j=t+1

cfj(1 + βP )j

,

and cft denotes the expected cash flow at time t, and βP is the risk-adjusted dis-count rate (or required rate of return on the project).

Of course, this decision rule has to be reapplied every time new informationarrives during the deferral period to see how the optimal investment strategy mightchange in light of the new information.

It should be noted that the fact that real options are like financial options doesnot mean that they are the same. Real options are concerned about strategic deci-sions of a company, where degrees of freedom are limited to the capabilities of thecompany. In these strategic decisions different stakeholders play a role, especiallyif the resources needed for an investment are significant and thereby the continuityof the company is at stake. Real options therefore, always need to be seen in thelarger context of the company, whereas financial options can be used freely andindependently.

2 Possibilistic mean value and variance of fuzzy numbers

A fuzzy number A is a fuzzy set of the real line with a normal, (fuzzy) convexand continuous membership function of bounded support. The family of fuzzy

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numbers will be denoted by F . For any A ∈ F we shall use the notation [A]γ =[a1(γ), a2(γ)] for γ-level sets of A. Fuzzy numbers can also be considered aspossibility distributions [6]. If A ∈ F is a fuzzy number and x ∈ R a real numberthen A(x) can be interpreted as the degree of possibility of the statement ’x is A’.

Definition 2.1 A fuzzy set A ∈ F is called a trapezoidal fuzzy number with core[a, b], left width α and right width β if its membership function has the followingform

A(t) =

1−a− tα

if a− α ≤ t < a

1 if a ≤ t ≤ b

1−t− bβ

if b < t ≤ b+ β

0 otherwise

and we use the notation A = (a, b, α, β). It can easily be shown that

[A]γ = [a− (1− γ)α, b+ (1− γ)β], ∀γ ∈ [0, 1].

The support of A is (a− α, b+ β).

A trapezoidal fuzzy number with core [a, b] may be seen as a context-dependentdescription (α and β define the context) of the property ”the value of a real vari-able is approximately in [a, b]”. If A(t) = 1 then t belongs to A with degree ofmembership one (i.e. a ≤ t ≤ b), and if A(t) = 0 then t belongs to A with degreeof membership zero (i.e. t /∈ (a− α, b+ β), t is too far from [a, b]), and finally if0 < A(t) < 1 then t belongs to A with an intermediate degree of membership (i.e.t is close enough to [a, b]). In a possibilistic setting A(t), t ∈ R, can be interpretedas the degree of possibility of the statement ”t is approximately in [a, b]”.

Let [A]γ = [a1(γ), a2(γ)] and [B]γ = [b1(γ), b2(γ)] be fuzzy numbers and letλ ∈ R be a real number. Using the extension principle we can verify the followingrules for addition and scalar muliplication of fuzzy numbers

[A+B]γ = [a1(γ) + b1(γ), a2(γ) + b2(γ)], [λA]γ = λ[A]γ . (4)

Let A ∈ F be a fuzzy number with [A]γ = [a1(γ), a2(γ)], γ ∈ [0, 1]. In [5] weintroduced the (crisp) possibilistic mean (or expected) value of A as

E(A) =∫ 1

0γ(a1(γ) + a2(γ))dγ =

∫ 1

0γ · a1(γ) + a2(γ)

2dγ∫ 1

0γ dγ

,

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i.e., E(A) is nothing else but the level-weighted average of the arithmetic meansof all γ-level sets, that is, the weight of the arithmetic mean of a1(γ) and a2(γ) isjust γ. It can easily be proved that E : F → R is a linear function (with respect tooperations (4)). In [5] we also introduced the (possibilistic) variance of A ∈ F as

σ2(A) =∫ 1

0γ

([a1(γ) + a2(γ)

2− a1(γ)

]2

+[a1(γ) + a2(γ)

2− a2(γ)

]2)dγ

=12

∫ 1

0γ(a2(γ)− a1(γ)

)2dγ.

i.e. the possibilistic variance of A is defined as the expected value of the squareddeviations between the arithmetic mean and the endpoints of its level sets.

It is easy to see that if A = (a, b, α, β) is a trapezoidal fuzzy number then

E(A) =∫ 1

0γ[a− (1− γ)α+ b+ (1− γ)β]dγ =

a+ b

2+β − α

6.

and

σ2(A) =(b− a)2

4+

(b− a)(α+ β)6

+(α+ β)2

24.

3 A hybrid approach to real option valuation

Usually, the present value of expected cash flows can not be be characterized by asingle number. However, our experiences with the Waeno research project on giga-investments show that managers are able to estimate the present value of expectedcash flows by using a trapezoidal possibility distribution of the form

S0 = (s1, s2, α, β),

i.e. the most possible values of the present value of expected cash flows lie in theinterval [s1, s2] (which is the core of the trapezoidal fuzzy number S0), and (s2+β)is the upward potential and (s1−α) is the downward potential for the present valueof expected cash flows.

In a similar manner one can estimate the expected costs by using a trapezoidalpossibility distribution of the form

X = (x1, x2, α′, β′),

i.e. the most possible values of expected cost lie in the interval [x1, x2] (which isthe core of the trapezoidal fuzzy number X), and (x2 + β′) is the upward potentialand (x1 − α′) is the downward potential for expected costs.

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Remark 3.1 The possibility distribution of expected costs and the present valueof expected cash flows could also be represented by nonlinear (e.g. Gaussian)membership functions. However, from a computational point of view it is easierto use linear membership functions and, more importantly, our experience showsthat senior managers prefer trapezoidal fuzzy numbers to Gaussian ones when theyestimate the uncertainties associated with future cash inflows and outflows.

In these circumstances we suggest the use of the following (heuristic) formulafor computing fuzzy real option values

FROV = S0e−δTN(d1)− Xe−rTN(d2), (5)

where,

d1 =ln(E(S0)/E(X)) + (r − δ + σ2/2)T

σ√T

, d2 = d1 − σ√T , (6)

and where, E(S0) denotes the possibilistic mean value of the present value ofexpected cash flows, E(X) stands for the the possibilistic mean value of expectedcosts and σ := σ(S0) is the possibilistic variance of the present value expected cashflows. Using formulas (4) for arithmetic operations on trapezoidal fuzzy numberswe find

FROV = (s1, s2, α, β)e−δTN(d1)− (x1, x2, α′, β′)e−rTN(d2) =

(s1e−δTN(d1)− x2e−rTN(d2), s2e−δTN(d1)− x1e

−rTN(d2),

αe−δTN(d1) + β′e−rTN(d2), βe−δTN(d1) + α′e−rTN(d2)).

(7)

Remark 3.2 It is clear that in (6) we can not use E(S0/X), because S0/X maynot be a fuzzy number.

Remark 3.3 The value of the fuzzy real option will also be a fuzzy number (beinga weighted difference of fuzzy numbers). What is then the relation between theprobabilistic real option value, computed by (2), and the fuzzy real option value,computed by (5)? It can easily be shown [9] that if the probabilistic density func-tion in (2) and the possibility distributions of S0 = (S0, 1) and X = (X, 1) are ofsymmetric triangular form then

ROV = E(FROV),

that is, the probabilistic real option value coincides with the possibilistic expectedvalue of the fuzzy real option. In the general case, possibility distributions do notsatisfy the properties of probabilistic density functions (and vice versa, probabilitydensity functions are mostly sub-normal), therefore ROV and E(FROV) can notbe compared (since one of them may not be defined).

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We have a specific context for the use of the real option valuation method withfuzzy numbers, which is the main motivation for our approach. Giga-investmentsrequire a basic investment exceeding 300 million euros and they normally have alife length of 15-25 years. The standard approach with the NPV or DCF methodsis to assume that uncertain revenues and costs associated with the investment canbe estimated as probabilistic values, which in turn are based on historic time seriesand observations of past revenues and costs.

It should be clear that the relevance of historic data diminishes very quicklyafter 2-3 years and that it is not worthwhile to claim that the time series have anypredictive value after 5 years (and even much more so for 15-25 years ahead). Inthe classical real options valuation methods this problem has been met by assumingthat either (i) the stock market is able to find a price for the impact of the investmenton the stock market value of the shares of the investing company (somehow doubt-ful for 15-25 years ahead), or that (ii) the uncertain revenues and costs are purelystochastic phenomena (described by, for instance, geometric Brownian motion).

We have discovered that giga-investments actually influence the end-user mar-kets in non-stochastic ways and that they are normally significant enough to havean impact on market strategies, on technology strategies, on competitive positionsand on business models. Thus, the use of assumptions on purely stochastic phe-nomena is not well-founded.

Example 3.1 Suppose we want to find a fuzzy real option value under the followingassumptions,

S0 = ($400 million, $600 million, $150 million, $150 million),

Figure 1: The possibility distribution of present values of expected cash flow.

r = 5% per year, T = 5 years, δ = 0.03 per year and

X = ($550 million, $650 million, $50 million, $50 million),

Figure 2: The possibility distribution of expected costs.

First calculate

σ(S0) =

√(s2 − s1)2

4+

(s2 − s1)(α+ β)6

+(α+ β)2

24= $154.11 million,

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Figure 3: The possibility distribution of real option values.

i.e. σ(S0) = 30.8%,

E(S0) =s1 + s2

2+β − α

6= $500 million,

and

E(X) =x1 + x2

2+β′ − α′

6= $600 million,

furthermore,

N(d1) = N

(ln(600/500) + (0.05− 0.03 + 0.3082/2)× 5

0.308×√

5

)= 0.589,

N(d2) = 0.321.

Thus, from (5) we obtain the fuzzy value of the real option as

FROV = ($40.15 million, $166.58 million, $88.56 million, $88.56 million).

The expected value of FROV is $103.37 million and its most possible valuesare bracketed by the interval

[$40.15 million, $166.58 million]

the downward potential (i.e. the maximal possible loss) is $48.41 million, and theupward potential (i.e. the maximal possible gain) is $255.15 million.

From Fig. 3 we can see that the set of most possible values of fuzzy real option[40.15, 166.58] is quite big. It follows from the huge uncertainties associated withcash inflows and outflows.

In the following we shall generalize the probabilistic decision rule for optimalinvestment strategy to a fuzzy setting: Where the maximum deferral time is T ,make the investment (exercise the option) at time t∗, 0 ≤ t∗ ≤ T , for which theoption, Ct∗ , attends its maximum value,

Ct∗ = maxt=0,1,...,T

Ct = Vte−δtN(d1)− Xe−rtN(d2), (8)

where

Vt = PV(cf0, . . . , cfT , βP )− PV(cf0, . . . , cft, βP ) = PV(cft+1, . . . , cfT , βP ),

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that is,

Vt = cf0 +T∑j=1

cfj(1 + βP )j

− cf0 −t∑

j=1

cfj(1 + βP )j

=T∑

j=t+1

cfj(1 + βP )j

,

where cft denotes the expected (fuzzy) cash flow at time t, βP is the risk-adjusteddiscount rate (or required rate of return on the project). However, to find a maxi-mizing element from the set

{C0, C1, . . . , CT },

is not an easy task because it involves ranking of trapezoidal fuzzy numbers.In our computerized implementation we have employed the following value

function to order fuzzy real option values, Ct = (cLt , cRt , αt, βt), of trapezoidal

form:

v(Ct) =cLt + cRt

2+ rA ·

βt − αt6

,

where rA ≥ 0 denotes the degree of the investor’s risk aversion. If rA = 0 then the(risk neutral) investor compares trapezoidal fuzzy numbers by comparing their pos-sibilistic expected values, i.e. he does not care about their downward and upwardpotentials.

4 Real options for strategic planning

There has been a suspicion for some time that capital invested in very large projects,with an expected life cycle of more than a decade is not very productive and thatthe overall activity around them is not very profitable. There is even some fear(which actually was articulated in the Waeno research program in Finland) thatgiga-investments in production facilities may consume capital, which could be usedmore effectively and more profitably in investment opportunities offered by theglobal financial markets. If the opportunities offered by a global market are used asreference points, arguments may be construed to show that long-term investmentsin production facilities will destroy capital. If this conclusion was to be made byrisk-taking investors, the consequences may be disastrous for major parts of thetraditional industry, which - by the way - forms the backbone of the economy andthe welfare of most industrial nations.

This discussion was real and urgent during the boom of the so-called Interneteconomy in 1998-2000, during which the market values of dotcom-start ups ex-ceeded the market values of multinational corporations running tens of pulp andpaper mills in a dozen countries.

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Giga-investments made in the paper- and pulp industry, in the heavy metal in-dustry and in other base industries, today face scenarios of slow (or even negative)growth (2-3 % p.a.) in their key markets and a growing over-capacity in Europe.The energy sector faces growing competition with lower prices and cyclic vari-ations of demand. There is also some statistics, which shows that productivityimprovements in these industries have slowed down to 1-2 % p.a., which opens theway for effective com-petitors to gain footholds in their main markets.

Giga-investments compete for major portions of the risk-taking capital, andas their life is long, compromises are made on their short-term productivity. Theshort-term productivity may not be high, as the life-long return of the investmentmay be calculated as very good. Another way of motivating a giga-investment is topoint to strategic advantages, which would not be possible without the investmentand thus will of-fer some indirect returns.

There are other issues. Global financial markets make sure that capital cannotbe used non-productively, as its owners are offered other opportunities and the cap-ital will move (often quite fast) to capture these opportunities. The capital markethas learned ’the American way’, i.e. there is a shareholder dominance among theactors, which has brought (often quite short-term) shareholder return to the fore-front as a key indicator of success, profitability and productivity.

There are lessons learned from the Japanese industry, which point to the impor-tance of immaterial investments. These lessons show that investments in buildings,production technology and supporting technology will be enhanced with immate-rial investments, and that these are even more important for reinvestments and forgradually growing maintenance investments.

The core products and services produced by giga-investments are enhancedwith life-time service, with gradually more advanced maintenance and financialadd-on services. These make it difficult to actually assess the productivity andprofitability of the original giga-investment, especially if the products and servicesare repositioned to serve other or emerging markets.

New technology and enhanced technological innovations will change the lifecycle of a giga-investment. The challenge is to find the right time and the rightinnovation to modify the life cycle in an optimal way. Technology providers areinvolved through-out the life cycle of a giga-investment, which should change theway in which we assess the profitability and the productivity of an investment.

Decision trees are excellent tools for making financial decisions where a lot ofvague information needs to be taken into account. They provide an effective struc-ture in which alternative decisions and the implications of taking those decisionscan be laid down and evaluated. They also help us to form an accurate, balancedpicture of the risks and rewards that can result from a particular choice.

In our empirical cases we have represented strategic planning problems by dy-

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Figure 4: The impact of 6 factors on the real option values. The (+/-) shows anincrease or a decrease of the ROV.

namic decision trees, in which the nodes are projects that can be deferred or post-poned for a certain period of time. Using the theory of real options we have beenable to identify the optimal path of the tree, i.e. the path with the biggest real optionvalue in the end of the planning period.

5 Nordic Telekom Inc.

The World’s telecommunications markets are undergoing a revolution. In the nextfew years mobile phones may become the World’s most common means of commu-nication, opening up new opportunities for systems and services. Characterized bylarge capital investment requirements under conditions of high regulatory, market,and technical uncertainty, the telecommunications industry faces many situationswhere strategic initiatives would benefit from real options analysis.

As the FROV method is applied to the telecom markets context and to thestrategic decisions of a telecom corporation we will have to understand in moredetail how the real option values are formed. In Fig. 4 the impact of a number offactors is summarized on the formation of the real option values.

The FROV will increase with an increasing volatility of cash flow estimates.The corporate management can be proactive and find (i) ways to expand to newmarkets, (ii) product innovations and (iii) (innovative) product combinations asend results of their strategic decisions. If the current value of expected cash flowswill increase, then the FROV will increase. A proactive management can influencethis by (for instance) developing market strategies or developing subcontractor re-lations.

The FROV will decrease if value is lost during the postponement of the invest-ment, but this can be countered by either creating business barriers for competitorsor by better managing key resources. An increase in risk-less returns will increasethe FROV, and this can be further enhanced by closely monitoring changes in theinterest rates. If the expected value of fixed costs goes up, the FROV will decreaseas opportunities of operating with less cost are lost. This can be countered by usingthe postponement period to explore and implement production scalability benefitsand/or to utilise learning benefits. The longer the time to maturity, the greater willbe the FROV. A proactive manage-ment can make sure of this development by(i) maintaining protective barriers, (ii) communicating implementation possibili-ties and (iii) maintaining a technological lead. The following example outlines the

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methodology used (to keep confidentiality we have modified the real setup) in thethe Nordic Telekom Inc. (NTI) case:

Example 5.1 Nordic Telekom Inc. is one of the most successful mobile communi-cations operators in Europe1 and has gained a reputation among its competitorsas a leader in quality, innovations in wireless technology and in building long-termcustomer relationships.

Still it does not have a dominating position in any of its customer segments,which is not even advisable in the European Common market, as there are always4-8 competitors with sizeable market shares. NTI would, nevertheless, like to havea position which would be dominant against any chosen competitor when definedfor all the markets in which NTI operates. NTI has associated companies thatprovide GSM services in five countries and one region: Finland, Norway, Sweden,Denmark, Estonia and the St. Petersburg region.

We consider strategic decisions for the planning period 2003-2011. There arethree possible alternatives for NTI: (i) introduction of third generation mobile so-lutions (3G); (ii) expanding its operations to other countries; and (iii) developingnew m-commerce solutions. The introduction of a 3G system can be postponed by amaximum of two years, the expansion may be delayed by maximum of one year andthe project on introduction of new m-commerce solutions should start immediately.

Our goal is to maximize the company’s cash flow at the end of the planning pe-riod (year 2011). In our computerized implementation we have represented NTI’sstrategic planning problem by a dynamic decision tree, in which the future expectedcash flows and costs are estimated by trapezoidal fuzzy numbers. Then using thetheory of fuzzy real options we have computed the real option values for all nodesof the dynamic decision tree. Then we have selected the path with the biggest realoption value in the end of the planning period (see Fig.5 - Fig.8).

Fig. 5. A simplified decision tree for Nordic telecom Inc.

Fig. 6 The optimal path: Introduction of 3G solutions in 2005 (after two years

1NTI is a fictional corporation.

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deferral time) and then proceed with 3G- Project 3 in 2009 (after four yearsdeferral time).

Fig. 7. Input data for the optimal (3G, 3G - Project 3) path.

Fig. 8. The fuzzy real option value of the 3G- Project 3.

6 Conclusions

Despite its appearance, the fuzzy real options model is quite practical and useful.Standard work in the field uses probability theory to account for the uncertaintiesinvolved in future cash flow estimates. This may be defended for financial op-tions, for which we can assume the existence of an efficient market with numerousplayers and numerous stocks for trading, which may justify the assumption of thevalidity of the laws of large numbers and thus the use of probability theory. Thesituation for real options is quite different. The option to postpone an investment(which in our case is a very large - so-called - giga-investment) will have conse-quences, differing from efficient markets, as the number of players producing theconsequences is quite small.

The imprecision we encounter when judging or estimating future cash flowsis not stochastic in nature, and the use of probability theory gives us a misleadinglevel of precision and a notion that consequences somehow are repetitive. Thisis not the case, the uncertainty is genuine, i.e. we simply do not know the exactlevels of future cash flows. Without introducing fuzzy real option models it wouldnot be possible to formulate this genuine uncertainty. The proposed model thatincorporates subjective judgments and statistical uncertainties may give investorsa better understanding of the problem when making investment decisions.

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[7] K. J. Leslie and M. P. Michaels, The real power of real options, The McKin-sey Quarterly, 3(1997) 5-22.

[8] T. A. Luehrman, Investment opportunities as real options: getting started onthe numbers, Harvard Business Review, July-August 1998, 51-67.

[9] P. Majlender, On probabilistic fuzzy numbers, in: Robert Trappl ed., Cyber-netics and Systems ’2003, Proceedings of the Sixteenth European Meetingon Cybernetics and Systems Research, Vienna, April 2-4, 2003, AustrianSociety for Cybernetic Studies, [ISBN 3-85206-160-1], 2002 520-523.

[10] R. Merton, Theory of rational option pricing, Bell Journal of Economics andManagement Science, 4(1973) 141-183.

[11] L.A. Zadeh, Fuzzy Sets, Information and Control, 8(1965) 338-353.

7 Citations

[A8] Christer Carlsson and Robert Fuller, A fuzzy approach to real option valu-ation, FUZZY SETS AND SYSTEMS, 139(2003) 297-312. [MR2006777]

in journals

A8-c63 Zdenek Zmeskal, Generalised soft binomial American real optionpricing model (fuzzystochastic approach), EUROPEAN JOURNAL OFOPERATIONAL RESEARCH, 207(2010), vol. 2, pp. 1096-1103.2010

15

http://dx.doi.org/10.1016/j.ejor.2010.05.045

Special attention in hybrid (fuzzystochastic) financial mod-elling is paid to option valuations, because of providing abun-dant and a lot of applications. Researchers has dealt with rel-atively new topic of fuzzystochastic option v aluation mod-els which are presented, e.g., in Zmeskal (2001), Simonelli(2001), Yoshida (2002), Carlsson and Fuller (2003), Yoshida(2003), Muzzioli and Torricelli (2004), Wu (2005), Cheng etal. (2005), Yoshida et al. (2005), Liyan Han and Wenli Chen(2006), Zhang et al. (2006), Wu (2007), Muzzioli and Rey-naerts (2007) and Thiagarajaha et al. (2007), Guerra et al.(2007), Chrysafis and Papadopoulos (2009). (page 1097)

A8-c62 Federica Cucchiella, Massimo Gastaldi, S C Lenny Koh, Perfor-mance improvement: an active life cycle product management, IN-TERNATIONAL JOURNAL OF SYSTEMS SCIENCE, 41(2010), is-sue 3, pp. 301-313. 2010http://dx.doi.org/10.1080/00207720903326886

A8-c61 Shu-Hsien Liao, Shiu-Hwei Ho, Investment project valuation basedon a fuzzy binomial approach, INFORMATION SCIENCES, 180(2010),issue 11, pp. 2124-2133. 2010http://dx.doi.org/10.1016/j.ins.2010.02.012

Carlsson and Fuller [A8] mentioned that the imprecision injudging or estimating future cash flows is not stochastic in na-ture, and that the use of the probability theory leads to a mis-leading level of precision. Their study introduced a (heuristic)real option rule in a fuzzy setting in which the present valuesof expected cash flows and expected costs are estimated bytrapezoidal fuzzy numbers. They determined the optimal ex-ercise time with the help of possibilistic mean value and vari-ance of fuzzy numbers. The proposed model that incorporatessubjective judgments and statistical uncertainties may give in-vestors a better understanding of the problem when makinginvestment decisions. (page 2125)

A8-c60 A. Cagri Tolga, C. Kahraman, M.L. Demircan, A comparativefuzzy real options valuation model using trinomial lattice and Black-Scholes approaches: A call center application, JOURNAL OF MULTIPLE-VALUED LOGIC AND SOFT COMPUTING, 16 (2010), No.1-2, pp.135-154. 2010

16

http://www.oldcitypublishing.com/FullText/MVLSCfulltext/MVLSC16.1-2fulltext/MVLSCv16n1-2p135-154Tolga.pdf

A8-c59 Xu Guoquan; Wang Yong, Real Options and Review of the energyfield, INQUIRY INTO ECONOMIC ISSUES, 1(2009), pp. 125-132(in Chinese). 2009

http://d.wanfangdata.com.cn/Periodical_jjwtts200901023.aspx

A8-c58 I. Ucal; C.Kahraman, Fuzzy real options valuation for oil invest-ments, TECHNOLOGICAL AND ECONOMIC DEVELOPMENT OFECONOMY, 15 (2009), issue 4, pp. 646-669. 2009http://dx.doi.org/10.3846/1392-8619.2009.15.646-669

Traditional valuation methods are less viable under uncer-tainty. Hence, other methods such as real options valuationmodels, which can minimize uncertainty, have become moreimportant. In this study, the hybrid approach suggested byCarlsson and Fuller is examined for the case of discrete com-pounding as this approach better models risky cash flows. Anew real options valuation model that will evaluate the invest-ment in a more realistic way is suggested by postponing thedefuzzification of parameters in early stages. The suggestedmodel has been applied to the data of an oil field investmentand in conclusion the loss of information caused by early-defuzzification has been determined. (page 646)

Carlsson and Fuller (2003) improved a fuzzy approach forreal options valuation, which is one of the mostly used realoptions valuation approaches, by applying a heuristic real op-tion rule in a fuzzy setting, where the present values of ex-pected cash flows and expected costs are estimated by trape-zoidal fuzzy numbers and they determined the optimal exer-cise time with the assistance of possibilistic mean value andvariance of fuzzy numbers. (page 650)

In this study we suggest a new model based on Carlsson &Fuller’s hybrid approach using discrete compounding insteadof continuous compounding by defuzzifying the costs andrevenues at a later stage than the based model and the basedmodel has been improved by fuzzifying interest rates and prob-abilities. (page 653)

17

Using real options valuation methods to analyse an invest-ment decision reduces the uncertainty to minimum and it en-sures that the investment assessment is made in as the mostrealistic way as possible. The model suggested by Carlssonand Fuller (2003) has been found to be the most frequentlyused method amongst the fuzzy real options assessment meth-ods during literature researches. In this model, however, it isobserved that the fuzzy revenue and expenditure values weredefuzzified at a relatively early stage. Early defuzzification ofthe fuzzy parameters causes information loss. In this study, anew model has been suggested; it postpones the defuzzifica-tion of fuzzy parameters in the real options valuation in orderto avoid this information loss. Carlsson and Fuller’s modelhas been re-arranged for discrete compounding and then thedefuzzification of revenue and expenditures has been post-poned and relavant equations have been formed for the casesof early defuzzifying probabilities and defuzzifying them atthe last stage. On the other hand, the difference between theapplications of this new model and of the early defuzzifyingmodel has been found to be the information loss due to theearly defuzzification. (pages 666-667)

A8-c57 Dan-mei Zhu, Tie Zhang, Xing-tong Wang, Dong-ling Chen, De-veloping an R&D projects portfolio selection decision system based onfuzzy logic, INTERNATIONAL JOURNAL OF MODELLING, IDEN-TIFICATION AND CONTROL, 8(2009), No. 3, pp. 205-212. 2009http://dx.doi.org/10.1504/IJMIC.2009.029265

A8-c56 Qian Wang, Keith Hipel, and Marc Kilgour, Fuzzy Real Optionsin Brownfield Redevelopment Evaluation, JOURNAL OF APPLIEDMATHEMATICS AND DECISION SCIENCES, Volume 2009(2009),Article ID 817137, 19 pages. 2009http://dx.doi.org/10.1155/2009/817137

The methods proposed to overcome the problem of privaterisk include utility theory [1, 7], the Integrated Value Process[4], Monte-Carlo simulation [8, 9], and unique and privaterisks [10]. This paper uses fuzzy real options, developed byCarlsson and Fuller [A8], to represent private risk. Represent-ing private risks by fuzzy variables leaves room for informa-tion other than market prices, such as expert experience andsubjective estimation, to be taken into account. In addition,

18

the model of Carlsson and Fuller can be generalized to allowparameters other than present value and exercise price [A8]to be fuzzy variables, utilizing the transformationmethod ofHanss [12]. (page 2)

Carlsson and Fuller assume that the present value and exer-cise price in the option formula are fuzzy variables with trape-zoidal fuzzy membership functions. Because inputs includefuzzy numbers, the value of a fuzzy real options is a fuzzynumber as well. A final crisp Real Option Value (ROV) canbe calculated as the expected value of the fuzzy ROV [A8].

...The concise and effective fuzzy real options approach pro-posed by Carlsson and Fuller is widely applicable; see [19-21]. (page 4)

A8-c55 Mohammad Modarres; Farhad Hassanzadeh, A Robust Optimiza-tion Approach to R&D Project Selection, WORLD APPLIED SCI-ENCES JOURNAL, 7(2009), No. 5, pp. 582-592. 2009

http://idosi.org/wasj/wasj7%285%29/4.pdf

A8-c54 Y.-Q. Xia, J.-F. Chen, Fuzzy optimization of real options valu-ation for multi-phase R&D project, Shanghai Jiaotong Daxue Xue-bao/Journal of Shanghai Jiaotong University, 43(2009), pp. 583-586.2009

A8-c53 Konstantinos A. Chrysafis, Basil K. Papadopoulos, On theoreticalpricing of options with fuzzy estimators, JOURNAL OF COMPUTA-TIONAL AND APPLIED MATHEMATICS, 223 (2009), pp. 552-566.2009http://dx.doi.org/10.1016/j.cam.2007.12.006

Since the Black-Scholes option pricing formula is only ap-proximate, it leads to considerable errors, Trenev [17] ob-tained a refined formula for pricing options. In [19] Wu ap-plied fuzzy approach to the Black-Scholes formula. Zmeskal[20] applied the BlackScholes methodology of appraising eq-uity as a European call option. Carlsson and Fuller [A8] usedthe possibility theory for fuzzy real option valuation. (page552)

A8-c52 A. Thavaneswaran, S.S. Appadoo, A. Paseka, Weighted possibilis-tic moments of fuzzy numbers with applications to GARCH model-

19

ing and option pricing, MATHEMATICAL AND COMPUTER MOD-ELLING, 9(2009) 352-368. 2009http://dx.doi.org/10.1016/j.mcm.2008.07.035

Another issue is the non-synchronous record of an optionprice and the price of the underlying asset. To top it off, op-tions themselves are traded at bid/ask prices. Using modelsthat do not specifically account for these issues introduces er-rors - difference between theoretical and observed option pre-miums. In empirical literature examples of statistical model-ing of the errors include, among others, Jacquier and Jarrow[13] and Bakshi et al. [14]. We introduce an alternative char-acterization of the error. To this end, we model the uncertaintyabout the values of interest rate and stock price using fuzzynumbers. Along the same lines, we model the unobserved pa-rameter, volatility, as a fuzzy number. We replace the fuzzyinterest rate, the fuzzy stock price, and the fuzzy volatility bypossibilistic mean value in the fuzzy Black-Scholes formula.The initial stock price cannot be characterized by a singlenumber. Thus, we assume that the initial stock price is a fuzzynumber of the form S0 = (S1, S2, S3, S4). We use a fuzzynumber of the form e−rτ = (e−r4τ , e−r3τ , e−r2τ , ee−r1τ ) =[e−r4τ + γ(e−r3τ − e−r4τ ), e−r1τ − γ(e−r1τ − e−r2τ )] forthe discount factor and σ = (σ1, σ1, σ3, σ4) for the volatility.Given these definitions, we suggest the use of the followingfuzzy weighted possibilistic (heuristic) option formula as inCarlsson and Fuller [A8] for computing fuzzy option values.We take an example of the Black-Scholes formula for a divi-dend paying stock with exercise price K. (page 366)

A8-c51 DING Si-bo; HUANG Wei-lai; ZHANG Zi-gang Real Option Pric-ing Method Based on Cloud, SYSTEMS ENGINEERING, 26(2008),number 10, pp. 73-76 (in Chinese). 2008http://www.cqvip.com/qk/93285x/2008010/29012808.html

A8-c50 LI Yao-kuang; XIONG Xing-hua; XIA qiong; ZHANG Xing-yu,Fuzzy option pricing model for the evaluation of investees in venturecapital investment, JOURNAL OF HEFEI UNIVERSITY OF TECH-NOLOGY (NATURAL SCIENCE), 31(2008), number 9, pp. 1494-1496 (in Chinese). 2008

http://d.wanfangdata.com.cn/Periodical_hfgydxxb200809036.aspx

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A8-c49 A.C. Tolga and C. Kahraman, Fuzzy multiattribute evaluation ofR&D projects using a real options valuation model, INTERNATIONALJOURNAL OF INTELLIGENT SYSTEMS, 23(2008), pp. 1153-1176.2008http://dx.doi.org/10.1002/int.20312

In the literature, fuzzy real options valuation models havebeen developed under incomplete information. First, Carls-son and Fuller [A8] developed a heuristic real option valu-ation process in a fuzzy setting. In their study, present val-ues of expected costs and expected cash flows are calculatedby trapezoidal fuzzy numbers. They determined the optimalexercise time with the help of possibilistic mean value andvariance of fuzzy numbers. (page 1155)

A8-c48 Zhu, D.-M., Zhang, T., Chen, D.-L., Gao, H.-X., New fuzzy pricingapproach to real option, Dongbei Daxue Xuebao/Journal of Northeast-ern University, 29(2008), pp. 1544-1547. 2008

A8-c47 A.C. Tolga, Fuzzy multicriteria R&D project selection with a realoptions valuation model, JOURNAL OF INTELLIGENT AND FUZZYSYSTEMS, 19(2008), pp. 359-371. 2008

A8-c46 Onofre Martorell Cunill; Carles Mulet Forteza; Micaela RosselloMiralles, Valuing growth strategy management by hotel chains basedon the real options approach, TOURISM ECONOMICS, Volume 14,Number 3, September 2008 , pp. 511-526. 2008

A8-c45 Du, X.-J., Sun, S.-D., Si, S.-B., Cai, Z.-Q, Multi-objective R andD portfolio selection optimization under uncertainty, SYSTEM ENGI-NEERING THEORY AND PRACTICE, Volume 28, Issue 2, February2008, pp. 98-104. 2008

A8-c44 S.S. Appadoo, S.K. Bhatt, C.R. Bector, Application of possibilitytheory to investment decisions, FUZZY OPTIMIZATION AND DE-CISION MAKING, vol. 7, pp. 35-57. 2008http://dx.doi.org/10.1007/s10700-007-9023-9

A8-c43 A. Tiwari, K. Vergidis, Y. Kuo, Computer assisted decision makingfor new product introduction investments, COMPUTERS IN INDUS-TRY, 59(1), pp. 96-105. 2008http://dx.doi.org/10.1016/j.compind.2007.06.014

Real options are described as the means that allow the exten-sion of the decision making period for an investment, with-out binding to a particular choice [A8]. Real options prove a

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strategically life-saving approach for organisations that haveto deal with alternative choices on big investments regardingnew products or services. (page 97)

A8-c42 X.-J. Du, S.-D. Sun, J.-Q. Wang, J.-H. Hao, Multi-objective R andD portfolio selection optimization under uncertainty, Jisuanji JichengZhizao Xitong/Computer Integrated Manufacturing Systems, CIMS Vol-ume 13, Issue 9, September 2007, pp. 1826-1832+1846. 2007

A8-c41 Wang JT, Hwang WL, A fuzzy set approach for R & D portfolio se-lection using a real options valuation model, OMEGA-INTERNATIONALJOURNAL OF MANAGEMENT SCIENCE 35 (3): 247-257 JUN 2007http://dx.doi.org/10.1016/j.omega.2005.06.002

Carlsson and Fuller [A8] have developed a fuzzy real optionsmodel to evaluate a project that only considers a single op-tion. However, an R & D project usually involves multiplephases. Therefore, the compound options valuation modelthat involves options whose value is contingent on the valueof other options is more suitable to evaluate the R & D project.(pages 250-251)

The fuzzy value of the R & D project is defined as [A8,27]

V = Se−δT2M(a1, b1,

√T1/T2

)−C3e

−rT2M(a2, b2,

√T1/T2

)−C2e

−rT1N(a2)

(page 251)

A8-c40 Thavaneswaran A, Thiagarajah K, Appadoo SS Fuzzy coefficientvolatility (FCV) models with applications, MATHEMATICAL ANDCOMPUTER MODELLING, 45 (7-8): 777-786 APR 2007http://dx.doi.org/10.1016/j.mcm.2006.07.019

Following Carlsson and Fuller [B1, A8], higher order mo-ments of fuzzy numbers are defined. We also derive the mo-ments and possibilistic kurtosis of the proposed FCV models.(page 778)

A8-c39 Thiagarajah, K., Appadoo, S.S., Thavaneswaran, A. Option valua-tion model with adaptive fuzzy numbers, COMPUTERS AND MATH-EMATICS WITH APPLICATIONS, 53 (5), pp. 831-841. 2007http://dx.doi.org/10.1016/j.camwa.2007.01.011

Since the Black-Scholes option pricing formula is only ap-proximate, which leads to considerable errors, Trenev [8] ob-tained a refined formula for pricing options. Due to the fluc-tuation of the financial markets from time to time, some of the

22

input parameters in the Black-Scholes formula cannot alwaysbe expected in the precise sense. As a result, Wu [9] appliedfuzzy approach to the Black-Scholes formula. Zmeskal [10]applied Black-Scholes methodology of appraising equity asa European call option. He used the input data in a formof fuzzy numbers in his approach to option pricing. Carls-son and Fuller [A8] use possibility theory to fuzzy real optionvaluation. (page 832)

A8-c38 Yoshida, Y., Yasuda, M., Nakagami, J.-I., Kurano, M. A discrete-time American put option model with fuzziness of stock prices, FUZZYOPTIMIZATION AND DECISION MAKING, 4 (3), pp. 191-207.2005http://dx.doi.org/10.1007/s10700-005-1889-9

The option pricing for stocks plays an important role in stockmarkets. Approaches with fuzzy logic to European optionsare discussed by some authors (Carlsson and Fuller (2003),Simonelli (2001), Yoshida (2003), Zmeskal (2001)). (page191)

A8-c37 Wang, G., Wu, C. The integral over a directed line segment of fuzzymapping and its applications, INTERNATIONAL JOURNAL OF UN-CERTAINTY FUZZINESS AND KNOWLEDGE-BASED SYSTEMS,12 (4), pp. 543-556. 2004http://dx.doi.org/10.1142/S0218488504002977

in proceedings and inedited volumes

A8-c27 Sibo Ding, Valuation of Reverse Logistics Company Based on FROand FMADM, in: 2nd International Conference on e-Business and In-formation System Security (EBISS), Wuhan, China, May 22-23, 2010,pp. 1-4. 2010http://dx.doi.org/10.1109/EBISS.2010.5473406

Black and Scholes [1] succeeded in establishing a option pric-ing formula and promoted the further development of finan-cial derivative markets. Dixit and Pindyck [2,3] focused onmulti-stage investment analysis, in which each stage must becompleted in the previous stage. They priced the project in-vestment opportunities at every stage of multi-phase, and fig-ured out the lowest cost of the next phase of the project. Carls-son and Fuller [A14, A8] studied the mean and variance of

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fuzzy number and applied them to the research of fuzzy realoption. Using fuzzy method, they assumed the present valueof expected cash flows and investment costs as fuzzy num-bers and employed fuzzy real option to investment decision-making . . . This paper developed the combination method thatuses fuzzy real option and fuzzy multiple attribute decisionmaking to evaluate the value of a reverse logistics company.(page 1)

A8-c26 Yibo Sun, Sameer Tilak, Ruppa K. Thulasiram, Kenneth Chiu,Markets, Mechanisms, Games, and Their Implications in Grids, in: Ra-jkumar Buyya, Kris Bubendorfer eds., Market-Oriented Grid and Util-ity Computing, Wiley Series on Parallel and Distributed Computing,Wiley & Sons, [ISBN 978-0-470-28768-2], pp. 29-48. 2010

Carlsson and Fuller [A8] apply a hybrid approach to valuingreal options. (page 46)

A8-c25 Maria Letizia Guerra, Laerte Sorini, Luciano Stefanini, Value Func-tion Computation in Fuzzy Real Options by Differential Evolution,Ninth International Conference on Intelligent Systems Design and Ap-plications, November 30-December 02, 2009, Pisa, Italy, [ISBN 978-0-7695-3872-3], pp. 324-329. 2009http://dx.doi.org/10.1109/ISDA.2009.232

We present a real options that will be evaluated within a fuzzysetting; more specifically, the present values of expected cashflows and expected costs are estimated by fuzzy numbers. Tothe best of our knowledge, such an approach has never beendiscussed in the literature, with the exception of Carlsson andFuller [A8], that interpret the possibility of making an invest-ment decision in terms of a European option, while we use anAmerican option. (page 324)

A8-c24 Wei Li, Liyan Han, The fuzzy binomial option pricing model un-der Knightian uncertainty, 6th International Conference on Fuzzy Sys-tems and Knowledge Discovery, FSKD 2009 Volume 4, [ISBN 978-076953735-1], 14-16, August, Tianjin, China, Article number 5359195,pp. 399-403. 2009http://dx.doi.org/10.1109/FSKD.2009.252

A8-c23 Miao Jing-yi; Sun Zhen-hua; Miao Miao, The Technological In-vestment Decision Study Based on Fuzzy Real Option, Proceedings -

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2009 International Symposium on Information Engineering and Elec-tronic Commerce, IEEC 2009, 16-17 May, 2009, art. no. 5175103, pp.202-206. 2009http://dx.doi.org/10.1109/IEEC.2009.47

Therefore, fuzzy set theory plays an important role in invest-ment decision of project. As a result, we combine the theoryof the fuzzy set and B-S equation as the basis of further stud-ies [A8], set up one corresponding fuzzy real option assess-ment model, and develop the calculation procedures using theassessment model to evaluate real option. (page 203)

A8-c22 Yibo Sun, Sameer Tilak, Ruppa K. Thulasiram, and Kenneth Chiu,Markets, Mechanisms, Games, and Their Implications in Grids, in: Ra-jkumar Buyya, Kris Bubendorfer eds., Market-Oriented Grid and Util-ity Computing, Wiley, [ISBN: 978-0-470-28768-2], pp. 29-48. 2009

A8-c21 Qian Wang, Keith W. Hipel, D. Marc Kilgour, Solving the PrivateRisk Problem in Brownfield Redevelopment using Fuzzy Real Options,Proceedings of GDN 2009: International Conference on Group Deci-sion and Negotiation Toronto, Canada, pp. 178-180. 2009

This proposed approach is based on Carlsson and Fuller’s ar-ticle on the fuzzy real options [A8]. But since the private riskis usually reflected as the volatility (σ), the transformationmethod developed by Hanss [5] is integrated in order to beable to make σ a fuzzy variable. (page 179)

A8-c20 Zhengbiao Qin, Yun Pu, and Minjie Hu, Investment Decisions inthe BOT Transport Infrastructure Applying Fuzzy Real Option, in: Pro-ceedings of the 8th International Conference of Chinese Logistics andTransportation Professionals - Logistics: The Emerging Frontiers ofTransportation and Development in China, pp. 1584-1589. 2008http://dx.doi.org/10.1061/40996(330)231

A8-c19 D. Allenotor, R.K. Thulasiram, Grid resources pricing: A novelfinancial option based quality of service-profit quasi-static equilibriummodel, 9th IEEE/ACM International Conference on Grid Computing,pp. 75-84. 2008http://dx.doi.org/10.1109/GRID.2008.4662785

Carlsson and Fuller in [A8] apply a hybrid approach to valu-ing real options. Their method incorporates real option, fuzzylogic, and probability to account for the uncertainty involved

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in the valuation of future cash flow estimates. The resultsof the research efforts given in [A8] and [18] have no for-mal reference to the QoS that characterize a decision system.Carlsson and Fuller [A8] apply fuzzy methods to measurethe level of decision uncertainties and did not price grid re-sources. (pages 76-77)

A8-c18 A.C. Tolga, C. Kahraman, Fuzzy multi-criteria evaluation of R&Dprojects and a fuzzy trinomial lattice approach for real options, in: Pro-ceedings of the 3rd International Conference on Intelligent System andKnowledge Engineering, ISKE 2008, November 17-19, 2008, Xiamen,China, Article number 4730966, pp. 418-423. 2008http://dx.doi.org/10.1109/ISKE.2008.4730966

Real options give a right but not an obligation to make ornot to make an investment for a certain period. For instance,organizing a Research and Development (R&D) laboratoryinvestment gives the company right to research and developnew products now but not the obligation at future. Real op-tions were first introduced by Trigeorgis [2]. Huchzermeierand Loch’s [3] model built real option for R&D managers asto when it is and when it is not worthwhile to delay commit-ments. The lack of past data and vague information drivesus to use fuzzy models. In the literature, fuzzy real optionsvaluation (FROV) models have been developed under incom-plete information. First, Carlsson and Fuller [A8] developeda heuristic real option valuation process in a fuzzy setting.In their study present values of expected costs and expectedcash flows are calculated by trapezoidal fuzzy numbers. (page418)

A8-c17 P. Majlender, Soft decision support systems for evaluating real andfinancial investments, in: Fuzzy Engineering Economics with Applica-tions, Studies in Fuzziness and Soft Computing series, 233/2008, pp.307-338. 2008

http://dx.doi.org/10.1007/978-3-540-70810-0_17

A8-c16 D. Kuchta, Optimization with fuzzy present worth analysis and ap-plications, in: Fuzzy Engineering Economics with Applications, Stud-ies in Fuzziness and Soft Computing, vol. 233/2008, Springer, [ISBN978-3-540-70809-4], pp. 43-69. 2008

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http://dx.doi.org/10.1007/978-3-540-70810-0_2

A8-c15 A. Dimitrovski, M. Matos, Fuzzy present worth analysis with cor-related and uncorrelated cash flows, in: Studies in Fuzziness and SoftComputing, vol. 233, pp. 11-41. 2008

http://dx.doi.org/10.1007/978-3-540-70810-0_2

A8-c14 David Allenotor, Ruppa K. Thulasiram, G-FRoM: Grid ResourcesPricing A Fuzzy Real Option Model In: IEEE International Conferenceon e-Science and Grid Computing, December 10-13, 2007, Bangalore,India, pp. 388-395. 2007http://dx.doi.org/10.1109/E-SCIENCE.2007.37

Current literature on real option approaches to valuing projectspresents real options framework in eight categories [15]: op-tion to defer, time-to-build option, option to alter, option toexpand, option to abandon, option to switch, growth options,and multiple integrating options. There are also efforts re-ported towards improving the selection and decision meth-ods used in the prediction of the capital that an investmentmay consume. Carlsson and Fuller in [A8] apply a hybridapproach to valuing real options. Their method incorporatesreal option and fuzzy logic and some aspects of probabilityto account for the uncertainty involved in the valuation of fu-ture cash ow estimates. The results of the research effortsgiven in [15] and [8] have no formal reference to the QoS thatcharacterize a decision system. Carlsson and Fuller [8] applyfuzzy methods to measure the level of decision uncertainties,however, there is a lack of indication on how accurate the de-cisions could be. (page 389)

A8-c13 Cheng, Jao-Hong Lee, Chen-Yu , Product Outsourcing under Un-certainty: an Application of Fuzzy Real Option Approach, IEEE Inter-national Fuzzy Systems Conference, 2007 (FUZZ-IEEE 2007), 23-26July 2007, London, [ISBN: 1-4244-1210-2], pp. 1-6. 2007http://dx.doi.org/10.1109/FUZZY.2007.4295675

Carlsson and Fuller [A8] stated that the relevance of historicdata diminished very quickly after 2-3 years, so it was notworthwhile to claim the predictive value of time series after 5years (and even much more so for 15-25 years ahead). (page2)

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A8-c12 Chen, Tao; Zeng, Yurong; Wang, Lin; Zhang, Jinlong, EvaluatingIT Investment Using a Hybrid Approach of Fuzzy Risk Analysis andReal Options, Fourth International Conference on Fuzzy Systems andKnowledge Discovery (FSKD 2007), Haikou, Hainan, China, 24-27Aug. 2007, vol.1, pp.135-139. 2007http://dx.doi.org/10.1109/FSKD.2007.276

A8-c11 David Allenotor and Ruppa K. Thulasiram, A Grid Resources Val-uation Model Using Fuzzy Real Option, in: Ivan Stojmenovic; RuppaK.Thulasiram; Laurence T.Yang; Weijia Jia; Minyi Guo; Rodrigo Fer-nandes de Mello eds., Parallel and Distributed Processing and Applica-tions, 5th International Symposium, ISPA2007, Niagara Falls, Canada,August 29-31, 2007, Lecture Notes in Computer Science, vol. 4742,Springer, pp. 622-632. 2007

http://dx.doi.org/10.1007/978-3-540-74742-0_56

Carlsson and Fuller in [A8] apply a hybrid approach (real op-tion, fuzzy logic, and probability theory) to value future op-tions. The results given in [2] and [A8] have no formal refer-ence to the QoS that characterize a decision system. Carlssonand Fuller [A8] apply fuzzy methods to measure the level ofdecision uncertainties, however, there is a lack of indicationon how accurate the decisions could be. (page 624)

A8-c10 Chen Tao, Zhang Jinlong, Yu Benhai, and Liu Shan A Fuzzy GroupDecision Approach to Real Option Valuation, in: Aijun An, Jerzy Ste-fanowski, Sheela Ramanna, Cory J. Butz, Witold Pedrycz, GuoyinWang (Eds.): Rough Sets, Fuzzy Sets, Data Mining and Granular Com-puting, 11th International Conference, RSFDGrC 2007, Toronto, Canada,May 14-16, 2007, Lecture Notes in Computer Science, Sublibrary:Lecture Notes in Artificial Intelligence, vol. 4482, Springer, [ISBN978-3-540-72529-9], pp. 103-110. 2007

http://dx.doi.org/10.1007/978-3-540-72530-5_12

A8-c9 Chen Tao, Zhang Jinlong, Liu Shan, and Yu Benhai, Fuzzy RealOption Analysis for IT Investment in Nuclear Power Station, Y. Shi etal. (Eds.): ICCS 2007, Part III, Lecture Notes in Computer Science,Volume 4489/2007, Springer, [ISBN 978-3-540-72587-9], pp. 953-959. 2007

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http://dx.doi.org/10.1007/978-3-540-72588-6_152

Step 5: the real option valuation of the investmentIn the last step, we can assess the real option value of the in-vestment based on the result obtained above. For the purposeof simplicity, we assume that only the expected payoff is un-certain and utilize the Black-Scholes pricing model. Then thefuzzy real option value of an investment is [A8]

FROV = V N(d1)−Xe−rTN(d2)

where

d1 =ln(E(V )/X) + (r + σ2/2)T

σ√T

, d2 = d1 − σ√T ,

Only V is a fuzzy number. E(V ) and σ represent, respec-tively, the possibilistic expected value and the standard devi-ation of fuzzy figure V . The computing result FROV is alsoa fuzzy number, representing the real option value of the in-vestment under consideration. (page 955)

A8-c8 Yoshida, Y. Option pricing theory in financial engineering from theviewpoint of fuzzy logic, in: Kahraman, Cengiz (Ed.) Fuzzy Applica-tions in Industrial Engineering, Series: Studies in Fuzziness and SoftComputing , Vol. 201, pp. 229-243. 2006

http://dx.doi.org/10.1007/3-540-33517-X_8

A8-c7 Zhang JL, Du HB, Tang WS, Pricing R & D option with combin-ing randomness and fuzziness, in: Computational Intelligence, Inter-national Conference on Intelligent Computing, ICIC 2006, Kunming,China, August 16-19, 2006. Proceedings, Part II, LECTURE NOTESIN ARTIFICIAL INTELLIGENCE, vol. 4114, pp. 798-808. 2006

http://dx.doi.org/10.1007/11816171_100

Carlsson and Fuller [A8] gianed the present value of expectedcash flow could not usually be characterized by a single num-ber. However, their experiences with the practical projectsshowed that managers were able to estimate the present valueof expected cash flows by using a trapezoidal fuzzy variable.(page 799)

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A8-c6 Karsak EE, A generalized fuzzy optimization framework for R &D project selection using real options valuation, in: ComputationalScience and Its Applications - ICCSA 2006, LECTURE NOTES INCOMPUTER SCIENCE 3982: 918-927 2006

http://dx.doi.org/10.1007/11751595_96

Lately, options valuation approach has been proposed as amore suitable alternative for determining the benefits fromR&D projects [9]. Options approach deviates from the con-ventional DCF approach in that it views future investment op-portunities as rights without obligations to take some actionin the future [3]. The asymmetry in the options expands theNPV to include a premium beyond the static NPV calculation,and thus presumably increase the total value of the project andthe probability of justification. Carlsson and Fuller [A8] fur-ther extended the use of options approach in R&D project val-uation by considering the possibilistic mean and variance offuzzy cash flow estimates. This paper presents a novel fuzzyformulation for R&D project selection accounting for projectinteractions with the objective of maximizing the net bene-fit based on expanded net present value which incorporatesthe real options inherent in R&D projects in a fuzzy setting.Although a fuzzy optimization model is provided for R&Dportfolio selection in [14], the project interactions are com-pletely ignored. Compared with the real options valuationprocedures delineated in [A8, 14], the valuation approach uti-lized in this paper models exercise price as a stochastic vari-able enabling to deal with technological uncertainties in realoptions analysis, and considers both the benefits of keepingthe development option alive and the opportunity cost of de-laying development. (page 919)

A8-c5 T. Sato, S. Takahashi, C, Huang and H. Inoue, Option pricing withfuzzy barrier conditions, in: Y. Liu, G. Chen and M. Ying eds., Pro-ceedings of the Eleventh International Fuzzy systems Association WorldCongress, July 28-31, 2005, Beijing, China, 2005 Tsinghua UniversityPress and Springer, [ISBN 7-302-11377-7] pp. 380-384. 2005

A8-c4 Garcia, F.A.A., Fuzzy real option valuation in a power station reengi-neering project, Soft Computing with Industrial Applications - Pro-ceedings of the Sixth Biannual World Automation Congress, [ISBN

30

1-889335-21-5], pp. 281-287. 2004

http://ieeexplore.ieee.org/xpls/abs_all.jsp?arnumber=1439379

in books

A8-c1 Ketty Peeva; Yordan Kyosev, Fuzzy relational calculus: Theory, ap-plications and software, Advances in Fuzzy Systems - Applications andTheory, Vol. 22, World Scientific. 2004

in Ph.D. disserations

• Markku Heikkila, R&D investment decisions with real options - Prof-itability and Decision Support, Abo Akademi University, Abo, Finland,[ISBN 978-952-12-2379-2]. 2009

• Peter Majlender, A Normative Approach to Possibility Theory and SoftDecision Support, Turku Centre for Computer Science, Institute forAdvanced Management Systems Research, Abo Akademi University,No 54, [ISBN 952-12-1409-0]. 2004

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