A fuzzy approach to real option valuation ∗
Christer Carlsson
christer.carlsson@abo.fi
Robert Fullér
rfuller@abo.fi
Abstract
To have a real option means to have the possibility for a certain period to
either choose for or against making an invetsment decision, without binding
oneself up front. The real option rule is that one should invest today only if
the net present value is high enough to compensate for giving up the value
of the option to wait. Because the option to invest loses its value when the
investment is irreversibly made, this loss is an opportunity cost of investing.
The main question that a management group must answer for a deferrable investment opportunity is: How long do we postpone the investment, if we can
postpone it, up to T time periods? In this paper we shall introduce a (heuristic) real option rule in a fuzzy setting, where the present values of expected
cash flows and expected costs are estimated by trapezoidal fuzzy numbers.
We shall determine the optimal exercise time by the help of possibilistic mean
value and variance of fuzzy numbers.
1
Probabilistic real option valuation
Options are known from the financial world where they represent the right to buy
or sell a financial value, mostly a stock, for a predetermined price (the exercise
price), without having the obligation to do so. The actual selling or buying of the
underlying value for the predetermined price is called exercising your option. One
would only exercise the option if the underlying value is higher than the exercise
price in case of a call option (the right to buy) or lower than the exercise prise in
the case of a put option (the right to sell). In 1973 Black and Scholes [4] made
a major breakthrough by deriving a differential equation that must be satisfied by
the price of any derivative security dependent on a non-dividend paying stock. For
risk-neutral investors the Black-Scholes pricing formula for a call option is
C0 = S0 N (d1 ) − Xe−rT N (d2 ),
∗
The final version of this paper appeared in: Fuzzy Sets and Systems, 139(2003) 297-312.
1
where
d1 =
ln(S0 /X) + (r + σ 2 /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 probability that a random draw from a standard normal distribution will be less than
d, X is the exercise price, r is the annualized continuously compounded rate on a
safe asset with the same maturity as the expiration of the option, T is the time to
maturity of the option (in years) and σ denotes the standard deviation of the annualized 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 = S0 e−δT N (d1 ) − Xe−rT N (d2 )
where,
(1)
√
ln(S0 /X) + (r − δ + σ 2 /2)T
√
, d2 = d1 − σ T
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 financial
ones [1]. To have a ”real option” means to have the possibility for a certain period
to either choose for or against making an invetsment decision, without binding
oneself 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 been
developed for financial options, which is quite different from traditional discounted
cashflow investment approaches. In traditional investment approaches investments
activities or projects are often seen as now or never and the main question is
whether 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 uncertainty about the exact outcome of the investment. To help with these tough
decisions valuation methods as Net Present Value (NPV) or Discounted Cash Flow
(DCF) have been developed. And since these methods ignore the value of flexibility and discount heavily for external uncertainty involved, many interesting and
innovative activities and projects are cancelled because of the uncertainties.
However, only a few investment projects are now or never. Often it is possible
to delay, modify or split up the project in strategic components which generate
important learning effects (and therefore reduce uncertainty). And in those cases
option 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 enough
to compensate for giving up the value of the option to wait. Because the option
d1 =
2
to invest loses its value when the investment is irreversibly made, this loss is an
opportunity cost of investing. Following Leslie and Michaels [7] we will compute
the value of a real option by
ROV = S0 e−δT N (d1 ) − Xe−rT N (d2 )
where,
(2)
√
ln(S0 /X) + (r − δ + σ 2 /2)T
√
, d2 = d1 − σ T
σ T
and where ROV denotes the current real option value, S0 is the present value of
expected cash flows, X is the (nominal) value of fixed costs, σ quantifies the uncertainty of expected cash flows, and δ denotes the value lost over the duration of
the option.
We illustrate the main difference between the traditional (passive) NPV decision rule and the (active) real option approach by an example quoted from ([7],
page 10):
d1 =
. . . another oil company has the opportunity to acquire a five-year licence on block. When developed, the block is expected to yield 50
million barrels of oil. The current price of a barell of oil from this
field is $10 and the present value of the development costs is $600
million. 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 the
opportunity. But what would option valuation make of the same case?
To begin with, such a valuation would recognize the importance of uncertainty, which the NPV analysis effectively assumes away. There are
two major sources of uncertainty affecting the value of the block: the
quantity and the price of the oil. One can make a reasonable estimate
of the quantity of the oil by analyzing historical exploration data in
geologically similar areas. Similarly, historical data on the variablity
of oil prices is readily available.
Assume for the sake of argument that these two sources of uncertainty
jointly result in a 30 percent standard deviation (σ) around the growth
rate of the value of operating cash inflows. Holding the option also
obliges one to incur the annual fixed costs of keeping the reserve active
- let us say, $15 million. This represents a dividend-like payout of
three percent (i.e. 15/500) of the value of the assets.
3
We already know that the duration of the option, T , is five years and
the 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 investment
opportunity is: How long do we postpone the investment up to T time periods? To
answer this question, Benaroch and Kauffman ([2], page 204) suggested the following decision rule for optimal investment strategy: Where the maximum deferral
time is T , make the investment (exercise the option) at time t∗ , 0 ≤ t∗ ≤ T , for
which the option, Ct∗ , is positive and attends its maximum value,
Ct∗ =
max Ct = Vt e−δt N (d1 ) − Xe−rt N (d2 ),
t=0,1,...,T
(3)
where
Vt = PV(cf 0 , . . . , cf T , βP ) − PV(cf 0 , . . . , cf t , βP ) = PV(cf t+1 , . . . , cf T , βP ),
that is,
Vt = cf 0 +
T
X
j=1
t
T
X
X
cf j
cf j
cf j
−
cf
−
=
,
0
j
j
(1 + βP )
(1 + βP )
(1 + βP )j
j=1
j=t+1
and cf t denotes the expected cash flow at time t, and βP is the risk-adjusted discount rate (or required rate of return on the project).
Of course, this decision rule has to be reapplied every time new information
arrives during the deferral period to see how the optimal investment strategy might
change in light of the new information.
It should be noted that the fact that real options are like financial options does
not mean that they are the same. Real options are concerned about strategic decisions of a company, where degrees of freedom are limited to the capabilities of the
company. In these strategic decisions different stakeholders play a role, especially
if the resources needed for an investment are significant and thereby the continuity
of the company is at stake. Real options therefore, always need to be seen in the
larger context of the company, whereas financial options can be used freely and
independently.
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) convex
and continuous membership function of bounded support. The family of fuzzy
4
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 as
possibility distributions [6]. If A ∈ F is a fuzzy number and x ∈ R a real number
then 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 following
form

a−t



1−
if a − α ≤ t < a


α

 1
if a ≤ t ≤ b
A(t) =
t−b



if b < t ≤ b + β
1−


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-dependent
description (α and β define the context) of the property ”the value of a real variable is approximately in [a, b]”. If A(t) = 1 then t belongs to A with degree of
membership one (i.e. a ≤ t ≤ b), and if A(t) = 0 then t belongs to A with degree
of membership zero (i.e. t ∈
/ (a − α, b + β), t is too far from [a, b]), and finally if
0 < 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 interpreted
as 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 let
λ ∈ R be a real number. Using the extension principle we can verify the following
rules 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] we
introduced the (crisp) possibilistic mean (or expected) value of A as
Z 1
a1 (γ) + a2 (γ)
Z 1
γ·
dγ
2
0
E(A) =
γ(a1 (γ) + a2 (γ))dγ =
,
Z 1
0
γ dγ
0
5
i.e., E(A) is nothing else but the level-weighted average of the arithmetic means
of all γ-level sets, that is, the weight of the arithmetic mean of a1 (γ) and a2 (γ) is
just γ. It can easily be proved that E : F → R is a linear function (with respect to
operations (4)). In [5] we also introduced the (possibilistic) variance of A ∈ F as
2 2 !
Z 1
a
(γ)
+
a
(γ)
a
(γ)
+
a
(γ)
1
2
1
2
σ 2 (A) =
γ
− a1 (γ) +
− a2 (γ)
dγ
2
2
0
Z
2
1 1
=
γ a2 (γ) − a1 (γ) dγ.
2 0
i.e. the possibilistic variance of A is defined as the expected value of the squared
deviations 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
Z 1
a+b β−α
E(A) =
+
.
γ[a − (1 − γ)α + b + (1 − γ)β]dγ =
2
6
0
and
σ 2 (A) =
3
(b − a)2 (b − a)(α + β) (α + β)2
+
+
.
4
6
24
A hybrid approach to real option valuation
Usually, the present value of expected cash flows can not be be characterized by a
single number. However, our experiences with the Waeno research project on gigainvestments show that managers are able to estimate the present value of expected
cash flows by using a trapezoidal possibility distribution of the form
S̃0 = (s1 , s2 , α, β),
i.e. the most possible values of the present value of expected cash flows lie in the
interval [s1 , s2 ] (which is the core of the trapezoidal fuzzy number S̃0 ), and (s2 +β)
is the upward potential and (s1 −α) is the downward potential for the present value
of expected cash flows.
In a similar manner one can estimate the expected costs by using a trapezoidal
possibility distribution of the form
X̃ = (x1 , x2 , α0 , β 0 ),
i.e. the most possible values of expected cost lie in the interval [x1 , x2 ] (which is
the core of the trapezoidal fuzzy number X̃), and (x2 + β 0 ) is the upward potential
and (x1 − α0 ) is the downward potential for expected costs.
6
Remark 3.1 The possibility distribution of expected costs and the present value
of expected cash flows could also be represented by nonlinear (e.g. Gaussian)
membership functions. However, from a computational point of view it is easier
to use linear membership functions and, more importantly, our experience shows
that senior managers prefer trapezoidal fuzzy numbers to Gaussian ones when they
estimate the uncertainties associated with future cash inflows and outflows.
In these circumstances we suggest the use of the following (heuristic) formula
for computing fuzzy real option values
FROV = S̃0 e−δT N (d1 ) − X̃e−rT N (d2 ),
(5)
where,
ln(E(S̃0 )/E(X̃)) + (r − δ + σ 2 /2)T
√
,
σ T
d1 =
√
d2 = d1 − σ T ,
(6)
and where, E(S̃0 ) denotes the possibilistic mean value of the present value of
expected cash flows, E(X̃) stands for the the possibilistic mean value of expected
costs and σ := σ(S̃0 ) is the possibilistic variance of the present value expected cash
flows. Using formulas (4) for arithmetic operations on trapezoidal fuzzy numbers
we find
FROV = (s1 , s2 , α, β)e−δT N (d1 ) − (x1 , x2 , α0 , β 0 )e−rT N (d2 ) =
(s1 e−δT N (d1 ) − x2 e−rT N (d2 ), s2 e−δT N (d1 ) − x1 e−rT N (d2 ),
−δT
αe
0 −rT
N (d1 ) + β e
N (d2 ), βe
−δT
0 −rT
N (d1 ) + α e
(7)
N (d2 )).
Remark 3.2 It is clear that in (6) we can not use E(S̃0 /X̃), because S̃0 /X̃ may
not be a fuzzy number.
Remark 3.3 The value of the fuzzy real option will also be a fuzzy number (being
a weighted difference of fuzzy numbers). What is then the relation between the
probabilistic 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 function in (2) and the possibility distributions of S̃0 = (S0 , 1) and X̃ = (X, 1) are of
symmetric triangular form then
ROV = E(FROV),
that is, the probabilistic real option value coincides with the possibilistic expected
value of the fuzzy real option. In the general case, possibility distributions do not
satisfy the properties of probabilistic density functions (and vice versa, probability
density functions are mostly sub-normal), therefore ROV and E(FROV) can not
be compared (since one of them may not be defined).
7
We have a specific context for the use of the real option valuation method with
fuzzy numbers, which is the main motivation for our approach. Giga-investments
require a basic investment exceeding 300 million euros and they normally have a
life length of 15-25 years. The standard approach with the NPV or DCF methods
is to assume that uncertain revenues and costs associated with the investment can
be estimated as probabilistic values, which in turn are based on historic time series
and observations of past revenues and costs.
It should be clear that the relevance of historic data diminishes very quickly
after 2-3 years and that it is not worthwhile to claim that the time series have any
predictive value after 5 years (and even much more so for 15-25 years ahead). In
the classical real options valuation methods this problem has been met by assuming
that either (i) the stock market is able to find a price for the impact of the investment
on the stock market value of the shares of the investing company (somehow doubtful for 15-25 years ahead), or that (ii) the uncertain revenues and costs are purely
stochastic phenomena (described by, for instance, geometric Brownian motion).
We have discovered that giga-investments actually influence the end-user markets in non-stochastic ways and that they are normally significant enough to have
an impact on market strategies, on technology strategies, on competitive positions
and on business models. Thus, the use of assumptions on purely stochastic phenomena is not well-founded.
Example 3.1 Suppose we want to find a fuzzy real option value under the following
assumptions,
S̃0 = ($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
X̃ = ($550 million, $650 million, $50 million, $50 million),
Figure 2: The possibility distribution of expected costs.
First calculate
s
(s2 − s1 )2 (s2 − s1 )(α + β) (α + β)2
σ(S̃0 ) =
+
+
= $154.11 million,
4
6
24
8
Figure 3: The possibility distribution of real option values.
i.e. σ(S̃0 ) = 30.8%,
E(S̃0 ) =
and
E(X̃) =
s1 + s2 β − α
+
= $500 million,
2
6
x1 + x2 β 0 − α 0
+
= $600 million,
2
6
furthermore,
ln(600/500) + (0.05 − 0.03 + 0.3082 /2) × 5
√
N (d1 ) = N
0.308 × 5
N (d2 ) = 0.321.
= 0.589,
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 values
are bracketed by the interval
[$40.15 million, $166.58 million]
the downward potential (i.e. the maximal possible loss) is $48.41 million, and the
upward 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 with
cash inflows and outflows.
In the following we shall generalize the probabilistic decision rule for optimal
investment 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 the
option, C̃t∗ , attends its maximum value,
C̃t∗ =
max C̃t = Ṽt e−δt N (d1 ) − X̃e−rt N (d2 ),
t=0,1,...,T
(8)
where
˜ 0 , . . . , cf
˜ T , βP ) − PV(cf
˜ 0 , . . . , cf
˜ t , βP ) = PV(cf
˜ t+1 , . . . , cf
˜ T , βP ),
Ṽt = PV(cf
9
that is,
˜0 +
Ṽt = cf
T
X
j=1
t
T
˜j
˜j
˜j
X
X
cf
cf
cf
˜
−
cf
−
=
,
0
(1 + βP )j
(1 + βP )j
(1 + βP )j
j=1
j=t+1
˜ t denotes the expected (fuzzy) cash flow at time t, βP is the risk-adjusted
where cf
discount rate (or required rate of return on the project). However, to find a maximizing element from the set
{C̃0 , C̃1 , . . . , C̃T },
is not an easy task because it involves ranking of trapezoidal fuzzy numbers.
In our computerized implementation we have employed the following value
R
function to order fuzzy real option values, C̃t = (cL
t , ct , αt , βt ), of trapezoidal
form:
R
cL
β t − αt
t + ct
v(C̃t ) =
+ rA ·
,
2
6
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 possibilistic expected values, i.e. he does not care about their downward and upward
potentials.
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 that
the overall activity around them is not very profitable. There is even some fear
(which actually was articulated in the Waeno research program in Finland) that
giga-investments in production facilities may consume capital, which could be used
more effectively and more profitably in investment opportunities offered by the
global financial markets. If the opportunities offered by a global market are used as
reference points, arguments may be construed to show that long-term investments
in production facilities will destroy capital. If this conclusion was to be made by
risk-taking investors, the consequences may be disastrous for major parts of the
traditional industry, which - by the way - forms the backbone of the economy and
the welfare of most industrial nations.
This discussion was real and urgent during the boom of the so-called Internet
economy in 1998-2000, during which the market values of dotcom-start ups exceeded the market values of multinational corporations running tens of pulp and
paper mills in a dozen countries.
10
Giga-investments made in the paper- and pulp industry, in the heavy metal industry 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 variations of demand. There is also some statistics, which shows that productivity
improvements in these industries have slowed down to 1-2 % p.a., which opens the
way for effective com-petitors to gain footholds in their main markets.
Giga-investments compete for major portions of the risk-taking capital, and
as their life is long, compromises are made on their short-term productivity. The
short-term productivity may not be high, as the life-long return of the investment
may be calculated as very good. Another way of motivating a giga-investment is to
point to strategic advantages, which would not be possible without the investment
and thus will of-fer some indirect returns.
There are other issues. Global financial markets make sure that capital cannot
be used non-productively, as its owners are offered other opportunities and the capital will move (often quite fast) to capture these opportunities. The capital market
has learned ’the American way’, i.e. there is a shareholder dominance among the
actors, which has brought (often quite short-term) shareholder return to the forefront as a key indicator of success, profitability and productivity.
There are lessons learned from the Japanese industry, which point to the importance of immaterial investments. These lessons show that investments in buildings,
production technology and supporting technology will be enhanced with immaterial investments, and that these are even more important for reinvestments and for
gradually growing maintenance investments.
The core products and services produced by giga-investments are enhanced
with life-time service, with gradually more advanced maintenance and financial
add-on services. These make it difficult to actually assess the productivity and
profitability of the original giga-investment, especially if the products and services
are repositioned to serve other or emerging markets.
New technology and enhanced technological innovations will change the life
cycle of a giga-investment. The challenge is to find the right time and the right
innovation to modify the life cycle in an optimal way. Technology providers are
involved through-out the life cycle of a giga-investment, which should change the
way in which we assess the profitability and the productivity of an investment.
Decision trees are excellent tools for making financial decisions where a lot of
vague information needs to be taken into account. They provide an effective structure in which alternative decisions and the implications of taking those decisions
can be laid down and evaluated. They also help us to form an accurate, balanced
picture of the risks and rewards that can result from a particular choice.
In our empirical cases we have represented strategic planning problems by dy11
Figure 4: The impact of 6 factors on the real option values. The (+/-) shows an
increase or a decrease of the ROV.
namic decision trees, in which the nodes are projects that can be deferred or postponed for a certain period of time. Using the theory of real options we have been
able to identify the optimal path of the tree, i.e. the path with the biggest real option
value in the end of the planning period.
5
Nordic Telekom Inc.
The World’s telecommunications markets are undergoing a revolution. In the next
few years mobile phones may become the World’s most common means of communication, opening up new opportunities for systems and services. Characterized by
large capital investment requirements under conditions of high regulatory, market,
and technical uncertainty, the telecommunications industry faces many situations
where strategic initiatives would benefit from real options analysis.
As the FROV method is applied to the telecom markets context and to the
strategic decisions of a telecom corporation we will have to understand in more
detail how the real option values are formed. In Fig. 4 the impact of a number of
factors 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 new
markets, (ii) product innovations and (iii) (innovative) product combinations as
end results of their strategic decisions. If the current value of expected cash flows
will increase, then the FROV will increase. A proactive management can influence
this by (for instance) developing market strategies or developing subcontractor relations.
The FROV will decrease if value is lost during the postponement of the investment, but this can be countered by either creating business barriers for competitors
or by better managing key resources. An increase in risk-less returns will increase
the FROV, and this can be further enhanced by closely monitoring changes in the
interest rates. If the expected value of fixed costs goes up, the FROV will decrease
as opportunities of operating with less cost are lost. This can be countered by using
the postponement period to explore and implement production scalability benefits
and/or to utilise learning benefits. The longer the time to maturity, the greater will
be the FROV. A proactive manage-ment can make sure of this development by
(i) maintaining protective barriers, (ii) communicating implementation possibilities and (iii) maintaining a technological lead. The following example outlines the
12
methodology used (to keep confidentiality we have modified the real setup) in the
the Nordic Telekom Inc. (NTI) case:
Example 5.1 Nordic Telekom Inc. is one of the most successful mobile communications operators in Europe1 and has gained a reputation among its competitors
as a leader in quality, innovations in wireless technology and in building long-term
customer 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 always
4-8 competitors with sizeable market shares. NTI would, nevertheless, like to have
a position which would be dominant against any chosen competitor when defined
for all the markets in which NTI operates. NTI has associated companies that
provide 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 are
three possible alternatives for NTI: (i) introduction of third generation mobile solutions (3G); (ii) expanding its operations to other countries; and (iii) developing
new m-commerce solutions. The introduction of a 3G system can be postponed by a
maximum of two years, the expansion may be delayed by maximum of one year and
the 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 period (year 2011). In our computerized implementation we have represented NTI’s
strategic planning problem by a dynamic decision tree, in which the future expected
cash flows and costs are estimated by trapezoidal fuzzy numbers. Then using the
theory of fuzzy real options we have computed the real option values for all nodes
of the dynamic decision tree. Then we have selected the path with the biggest real
option 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
1
NTI is a fictional corporation.
13
deferral time) and then proceed with 3G- Project 3 in 2009 (after four years
deferral 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 uncertainties
involved in future cash flow estimates. This may be defended for financial options, for which we can assume the existence of an efficient market with numerous
players and numerous stocks for trading, which may justify the assumption of the
validity of the laws of large numbers and thus the use of probability theory. The
situation 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 consequences, differing from efficient markets, as the number of players producing the
consequences is quite small.
The imprecision we encounter when judging or estimating future cash flows
is not stochastic in nature, and the use of probability theory gives us a misleading
level of precision and a notion that consequences somehow are repetitive. This
is not the case, the uncertainty is genuine, i.e. we simply do not know the exact
levels of future cash flows. Without introducing fuzzy real option models it would
not be possible to formulate this genuine uncertainty. The proposed model that
incorporates subjective judgments and statistical uncertainties may give investors
a better understanding of the problem when making investment decisions.
References
[1] James Alleman and Eli Noam eds., The New Investment Theory of Real Options and Its Implication for Telecommunications Economics, Kluwer Aca14
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[3] P. Balasubramanian, N. Kulatikala and J. Storck, Managing information
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[4] F. Black and M. Scholes, The pricing of options and corporate liabilities,
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numbers, Fuzzy Sets and Systems, 122(2001) 315-326.
[6] D.Dubois and H.Prade, Possibility Theory (Plenum Press, New York,1988).
[7] K. J. Leslie and M. P. Michaels, The real power of real options, The McKinsey Quarterly, 3(1997) 5-22.
[8] T. A. Luehrman, Investment opportunities as real options: getting started on
the numbers, Harvard Business Review, July-August 1998, 51-67.
[9] P. Majlender, On probabilistic fuzzy numbers, in: Robert Trappl ed., Cybernetics and Systems ’2003, Proceedings of the Sixteenth European Meeting
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[10] R. Merton, Theory of rational option pricing, Bell Journal of Economics and
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7
Citations
[A8] Christer Carlsson and Robert Fullér, A fuzzy approach to real option valuation, FUZZY SETS AND SYSTEMS, 139(2003) 297-312. [MR2006777]
in journals
A8-c63 Zdeněk Zmeškal, Generalised soft binomial American real option
pricing model (fuzzystochastic approach), EUROPEAN JOURNAL OF
OPERATIONAL 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 modelling is paid to option valuations, because of providing abundant and a lot of applications. Researchers has dealt with relatively new topic of fuzzystochastic option v aluation models 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 et
al. (2005), Yoshida et al. (2005), Liyan Han and Wenli Chen
(2006), Zhang et al. (2006), Wu (2007), Muzzioli and Reynaerts (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, Performance improvement: an active life cycle product management, INTERNATIONAL JOURNAL OF SYSTEMS SCIENCE, 41(2010), issue 3, pp. 301-313. 2010
http://dx.doi.org/10.1080/00207720903326886
A8-c61 Shu-Hsien Liao, Shiu-Hwei Ho, Investment project valuation based
on a fuzzy binomial approach, INFORMATION SCIENCES, 180(2010),
issue 11, pp. 2124-2133. 2010
http://dx.doi.org/10.1016/j.ins.2010.02.012
Carlsson and Fuller [A8] mentioned that the imprecision in
judging or estimating future cash flows is not stochastic in nature, and that the use of the probability theory leads to a misleading level of precision. Their study introduced a (heuristic)
real option rule in a fuzzy setting in which the present values
of expected cash flows and expected costs are estimated by
trapezoidal fuzzy numbers. They determined the optimal exercise time with the help of possibilistic mean value and variance of fuzzy numbers. The proposed model that incorporates
subjective judgments and statistical uncertainties may give investors a better understanding of the problem when making
investment decisions. (page 2125)
A8-c60 A. Çagri Tolga, C. Kahraman, M.L. Demircan, A comparative
fuzzy real options valuation model using trinomial lattice and BlackScholes approaches: A call center application, JOURNAL OF MULTIPLEVALUED LOGIC AND SOFT COMPUTING, 16 (2010), No.1-2, pp.
135-154. 2010
16
http://www.oldcitypublishing.com/FullText/MVLSCfulltext/MVLSC16.12fulltext/MVLSCv16n1-2p135-154Tolga.pdf
A8-c59 Xu Guoquan; Wang Yong, Real Options and Review of the energy
field, INQUIRY INTO ECONOMIC ISSUES, 1(2009), pp. 125-132
(in Chinese). 2009
http://d.wanfangdata.com.cn/Periodical_jjwtts200901023.aspx
A8-c58 I. Uçal; C.Kahraman, Fuzzy real options valuation for oil investments, TECHNOLOGICAL AND ECONOMIC DEVELOPMENT OF
ECONOMY, 15 (2009), issue 4, pp. 646-669. 2009
http://dx.doi.org/10.3846/1392-8619.2009.15.646-669
Traditional valuation methods are less viable under uncertainty. Hence, other methods such as real options valuation
models, which can minimize uncertainty, have become more
important. In this study, the hybrid approach suggested by
Carlsson and Fuller is examined for the case of discrete compounding as this approach better models risky cash flows. A
new real options valuation model that will evaluate the investment in a more realistic way is suggested by postponing the
defuzzification of parameters in early stages. The suggested
model has been applied to the data of an oil field investment
and in conclusion the loss of information caused by earlydefuzzification has been determined. (page 646)
Carlsson and Fuller (2003) improved a fuzzy approach for
real options valuation, which is one of the mostly used real
options valuation approaches, by applying a heuristic real option rule in a fuzzy setting, where the present values of expected cash flows and expected costs are estimated by trapezoidal fuzzy numbers and they determined the optimal exercise time with the assistance of possibilistic mean value and
variance of fuzzy numbers. (page 650)
In this study we suggest a new model based on Carlsson &
Fuller’s hybrid approach using discrete compounding instead
of continuous compounding by defuzzifying the costs and
revenues at a later stage than the based model and the based
model has been improved by fuzzifying interest rates and probabilities. (page 653)
17
Using real options valuation methods to analyse an investment decision reduces the uncertainty to minimum and it ensures that the investment assessment is made in as the most
realistic way as possible. The model suggested by Carlsson
and Fuller (2003) has been found to be the most frequently
used method amongst the fuzzy real options assessment methods during literature researches. In this model, however, it is
observed that the fuzzy revenue and expenditure values were
defuzzified at a relatively early stage. Early defuzzification of
the fuzzy parameters causes information loss. In this study, a
new model has been suggested; it postpones the defuzzification of fuzzy parameters in the real options valuation in order
to avoid this information loss. Carlsson and Fuller’s model
has been re-arranged for discrete compounding and then the
defuzzification of revenue and expenditures has been postponed and relavant equations have been formed for the cases
of early defuzzifying probabilities and defuzzifying them at
the last stage. On the other hand, the difference between the
applications of this new model and of the early defuzzifying
model has been found to be the information loss due to the
early defuzzification. (pages 666-667)
A8-c57 Dan-mei Zhu, Tie Zhang, Xing-tong Wang, Dong-ling Chen, Developing an R&D projects portfolio selection decision system based on
fuzzy logic, INTERNATIONAL JOURNAL OF MODELLING, IDENTIFICATION AND CONTROL, 8(2009), No. 3, pp. 205-212. 2009
http://dx.doi.org/10.1504/IJMIC.2009.029265
A8-c56 Qian Wang, Keith Hipel, and Marc Kilgour, Fuzzy Real Options
in Brownfield Redevelopment Evaluation, JOURNAL OF APPLIED
MATHEMATICS AND DECISION SCIENCES, Volume 2009(2009),
Article ID 817137, 19 pages. 2009
http://dx.doi.org/10.1155/2009/817137
The methods proposed to overcome the problem of private
risk include utility theory [1, 7], the Integrated Value Process
[4], Monte-Carlo simulation [8, 9], and unique and private
risks [10]. This paper uses fuzzy real options, developed by
Carlsson and Fullér [A8], to represent private risk. Representing private risks by fuzzy variables leaves room for information other than market prices, such as expert experience and
subjective estimation, to be taken into account. In addition,
18
the model of Carlsson and Fullér can be generalized to allow
parameters other than present value and exercise price [A8]
to be fuzzy variables, utilizing the transformationmethod of
Hanss [12]. (page 2)
Carlsson and Fullér assume that the present value and exercise price in the option formula are fuzzy variables with trapezoidal fuzzy membership functions. Because inputs include
fuzzy numbers, the value of a fuzzy real options is a fuzzy
number as well. A final crisp Real Option Value (ROV) can
be calculated as the expected value of the fuzzy ROV [A8].
..
.
The concise and effective fuzzy real options approach proposed by Carlsson and Fullér is widely applicable; see [1921]. (page 4)
A8-c55 Mohammad Modarres; Farhad Hassanzadeh, A Robust Optimization Approach to R&D Project Selection, WORLD APPLIED SCIENCES 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 valuation for multi-phase R&D project, Shanghai Jiaotong Daxue Xuebao/Journal of Shanghai Jiaotong University, 43(2009), pp. 583-586.
2009
A8-c53 Konstantinos A. Chrysafis, Basil K. Papadopoulos, On theoretical
pricing of options with fuzzy estimators, JOURNAL OF COMPUTATIONAL AND APPLIED MATHEMATICS, 223 (2009), pp. 552-566.
2009
http://dx.doi.org/10.1016/j.cam.2007.12.006
Since the Black-Scholes option pricing formula is only approximate, it leads to considerable errors, Trenev [17] obtained a refined formula for pricing options. In [19] Wu applied fuzzy approach to the Black-Scholes formula. Zmeskal
[20] applied the BlackScholes methodology of appraising equity as a European call option. Carlsson and Fullér [A8] used
the possibility theory for fuzzy real option valuation. (page
552)
A8-c52 A. Thavaneswaran, S.S. Appadoo, A. Paseka, Weighted possibilistic moments of fuzzy numbers with applications to GARCH model19
ing and option pricing, MATHEMATICAL AND COMPUTER MODELLING, 9(2009) 352-368. 2009
http://dx.doi.org/10.1016/j.mcm.2008.07.035
Another issue is the non-synchronous record of an option
price and the price of the underlying asset. To top it off, options themselves are traded at bid/ask prices. Using models
that do not specifically account for these issues introduces errors - difference between theoretical and observed option premiums. In empirical literature examples of statistical modeling of the errors include, among others, Jacquier and Jarrow
[13] and Bakshi et al. [14]. We introduce an alternative characterization of the error. To this end, we model the uncertainty
about the values of interest rate and stock price using fuzzy
numbers. Along the same lines, we model the unobserved parameter, volatility, as a fuzzy number. We replace the fuzzy
interest rate, the fuzzy stock price, and the fuzzy volatility by
possibilistic mean value in the fuzzy Black-Scholes formula.
The initial stock price cannot be characterized by a single
number. Thus, we assume that the initial stock price is a fuzzy
number of the form S0 = (S1 , S2 , S3 , S4 ). We use a fuzzy
number of the form e−r̃τ = (e−r4 τ , e−r3 τ , e−r2 τ , ee−r1 τ ) =
[e−r4 τ + γ(e−r3 τ − e−r4 τ ), e−r1 τ − γ(e−r1 τ − e−r2 τ )] for
the discount factor and σ = (σ1 , σ1 , σ3 , σ4 ) for the volatility.
Given these definitions, we suggest the use of the following
fuzzy weighted possibilistic (heuristic) option formula as in
Carlsson and Fullér [A8] for computing fuzzy option values.
We take an example of the Black-Scholes formula for a dividend paying stock with exercise price K. (page 366)
A8-c51 DING Si-bo; HUANG Wei-lai; ZHANG Zi-gang Real Option Pricing Method Based on Cloud, SYSTEMS ENGINEERING, 26(2008),
number 10, pp. 73-76 (in Chinese). 2008
http://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 venture
capital investment, JOURNAL OF HEFEI UNIVERSITY OF TECHNOLOGY (NATURAL SCIENCE), 31(2008), number 9, pp. 14941496 (in Chinese). 2008
http://d.wanfangdata.com.cn/Periodical_hfgydxxb200809036.aspx
20
A8-c49 A.C. Tolga and C. Kahraman, Fuzzy multiattribute evaluation of
R&D projects using a real options valuation model, INTERNATIONAL
JOURNAL OF INTELLIGENT SYSTEMS, 23(2008), pp. 1153-1176.
2008
http://dx.doi.org/10.1002/int.20312
In the literature, fuzzy real options valuation models have
been developed under incomplete information. First, Carlsson and Fullér [A8] developed a heuristic real option valuation process in a fuzzy setting. In their study, present values of expected costs and expected cash flows are calculated
by trapezoidal fuzzy numbers. They determined the optimal
exercise time with the help of possibilistic mean value and
variance of fuzzy numbers. (page 1155)
A8-c48 Zhu, D.-M., Zhang, T., Chen, D.-L., Gao, H.-X., New fuzzy pricing
approach to real option, Dongbei Daxue Xuebao/Journal of Northeastern University, 29(2008), pp. 1544-1547. 2008
A8-c47 A.C. Tolga, Fuzzy multicriteria R&D project selection with a real
options valuation model, JOURNAL OF INTELLIGENT AND FUZZY
SYSTEMS, 19(2008), pp. 359-371. 2008
A8-c46 Onofre Martorell Cunill; Carles Mulet Forteza; Micaela Rosselló
Miralles, Valuing growth strategy management by hotel chains based
on 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 and
D portfolio selection optimization under uncertainty, SYSTEM ENGINEERING THEORY AND PRACTICE, Volume 28, Issue 2, February
2008, pp. 98-104. 2008
A8-c44 S.S. Appadoo, S.K. Bhatt, C.R. Bector, Application of possibility
theory to investment decisions, FUZZY OPTIMIZATION AND DECISION MAKING, vol. 7, pp. 35-57. 2008
http://dx.doi.org/10.1007/s10700-007-9023-9
A8-c43 A. Tiwari, K. Vergidis, Y. Kuo, Computer assisted decision making
for new product introduction investments, COMPUTERS IN INDUSTRY, 59(1), pp. 96-105. 2008
http://dx.doi.org/10.1016/j.compind.2007.06.014
Real options are described as the means that allow the extension of the decision making period for an investment, without binding to a particular choice [A8]. Real options prove a
21
strategically life-saving approach for organisations that have
to deal with alternative choices on big investments regarding
new products or services. (page 97)
A8-c42 X.-J. Du, S.-D. Sun, J.-Q. Wang, J.-H. Hao, Multi-objective R and
D portfolio selection optimization under uncertainty, Jisuanji Jicheng
Zhizao Xitong/Computer Integrated Manufacturing Systems, CIMS Volume 13, Issue 9, September 2007, pp. 1826-1832+1846. 2007
A8-c41 Wang JT, Hwang WL, A fuzzy set approach for R & D portfolio selection using a real options valuation model, OMEGA-INTERNATIONAL
JOURNAL OF MANAGEMENT SCIENCE 35 (3): 247-257 JUN 2007
http://dx.doi.org/10.1016/j.omega.2005.06.002
Carlsson and Fullér [A8] have developed a fuzzy real options
model to evaluate a project that only considers a single option. However, an R & D project usually involves multiple
phases. Therefore, the compound options valuation model
that involves options whose value is contingent on the value
of 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]
p
p
Ṽ = S̃e−δT2 M a1 , b1 , T1 /T2 −C̃3 e−rT2 M a2 , b2 , T1 /T2 −C̃2 e−rT1 N (a2 )
(page 251)
A8-c40 Thavaneswaran A, Thiagarajah K, Appadoo SS Fuzzy coefficient
volatility (FCV) models with applications, MATHEMATICAL AND
COMPUTER MODELLING, 45 (7-8): 777-786 APR 2007
http://dx.doi.org/10.1016/j.mcm.2006.07.019
Following Carlsson and Fullér [B1, A8], higher order moments of fuzzy numbers are defined. We also derive the moments and possibilistic kurtosis of the proposed FCV models.
(page 778)
A8-c39 Thiagarajah, K., Appadoo, S.S., Thavaneswaran, A. Option valuation model with adaptive fuzzy numbers, COMPUTERS AND MATHEMATICS WITH APPLICATIONS, 53 (5), pp. 831-841. 2007
http://dx.doi.org/10.1016/j.camwa.2007.01.011
Since the Black-Scholes option pricing formula is only approximate, which leads to considerable errors, Trenev [8] obtained a refined formula for pricing options. Due to the fluctuation of the financial markets from time to time, some of the
22
input parameters in the Black-Scholes formula cannot always
be expected in the precise sense. As a result, Wu [9] applied
fuzzy approach to the Black-Scholes formula. Zmeskal [10]
applied Black-Scholes methodology of appraising equity as
a European call option. He used the input data in a form
of fuzzy numbers in his approach to option pricing. Carlsson and Fullér [A8] use possibility theory to fuzzy real option
valuation. (page 832)
A8-c38 Yoshida, Y., Yasuda, M., Nakagami, J.-I., Kurano, M. A discretetime American put option model with fuzziness of stock prices, FUZZY
OPTIMIZATION AND DECISION MAKING, 4 (3), pp. 191-207.
2005
http://dx.doi.org/10.1007/s10700-005-1889-9
The option pricing for stocks plays an important role in stock
markets. Approaches with fuzzy logic to European options
are discussed by some authors (Carlsson and Fullér (2003),
Simonelli (2001), Yoshida (2003), Zmeskal (2001)). (page
191)
A8-c37 Wang, G., Wu, C. The integral over a directed line segment of fuzzy
mapping and its applications, INTERNATIONAL JOURNAL OF UNCERTAINTY FUZZINESS AND KNOWLEDGE-BASED SYSTEMS,
12 (4), pp. 543-556. 2004
http://dx.doi.org/10.1142/S0218488504002977
in proceedings and in
edited volumes
A8-c27 Sibo Ding, Valuation of Reverse Logistics Company Based on FRO
and FMADM, in: 2nd International Conference on e-Business and Information System Security (EBISS), Wuhan, China, May 22-23, 2010,
pp. 1-4. 2010
http://dx.doi.org/10.1109/EBISS.2010.5473406
Black and Scholes [1] succeeded in establishing a option pricing formula and promoted the further development of financial derivative markets. Dixit and Pindyck [2,3] focused on
multi-stage investment analysis, in which each stage must be
completed in the previous stage. They priced the project investment opportunities at every stage of multi-phase, and figured out the lowest cost of the next phase of the project. Carlsson and Fullér [A14, A8] studied the mean and variance of
23
fuzzy number and applied them to the research of fuzzy real
option. Using fuzzy method, they assumed the present value
of expected cash flows and investment costs as fuzzy numbers and employed fuzzy real option to investment decisionmaking . . . This paper developed the combination method that
uses fuzzy real option and fuzzy multiple attribute decision
making 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: Rajkumar Buyya, Kris Bubendorfer eds., Market-Oriented Grid and Utility Computing, Wiley Series on Parallel and Distributed Computing,
Wiley & Sons, [ISBN 978-0-470-28768-2], pp. 29-48. 2010
Carlsson and Fullér [A8] apply a hybrid approach to valuing
real options. (page 46)
A8-c25 Maria Letizia Guerra, Laerte Sorini, Luciano Stefanini, Value Function Computation in Fuzzy Real Options by Differential Evolution,
Ninth International Conference on Intelligent Systems Design and Applications, November 30-December 02, 2009, Pisa, Italy, [ISBN 9780-7695-3872-3], pp. 324-329. 2009
http://dx.doi.org/10.1109/ISDA.2009.232
We present a real options that will be evaluated within a fuzzy
setting; more specifically, the present values of expected cash
flows and expected costs are estimated by fuzzy numbers. To
the best of our knowledge, such an approach has never been
discussed in the literature, with the exception of Carlsson and
Fullér [A8], that interpret the possibility of making an investment decision in terms of a European option, while we use an
American option. (page 324)
A8-c24 Wei Li, Liyan Han, The fuzzy binomial option pricing model under Knightian uncertainty, 6th International Conference on Fuzzy Systems and Knowledge Discovery, FSKD 2009 Volume 4, [ISBN 978076953735-1], 14-16, August, Tianjin, China, Article number 5359195,
pp. 399-403. 2009
http://dx.doi.org/10.1109/FSKD.2009.252
A8-c23 Miao Jing-yi; Sun Zhen-hua; Miao Miao, The Technological Investment Decision Study Based on Fuzzy Real Option, Proceedings -
24
2009 International Symposium on Information Engineering and Electronic Commerce, IEEC 2009, 16-17 May, 2009, art. no. 5175103, pp.
202-206. 2009
http://dx.doi.org/10.1109/IEEC.2009.47
Therefore, fuzzy set theory plays an important role in investment decision of project. As a result, we combine the theory
of the fuzzy set and B-S equation as the basis of further studies [A8], set up one corresponding fuzzy real option assessment model, and develop the calculation procedures using the
assessment 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: Rajkumar Buyya, Kris Bubendorfer eds., Market-Oriented Grid and Utility Computing, Wiley, [ISBN: 978-0-470-28768-2], pp. 29-48. 2009
A8-c21 Qian Wang, Keith W. Hipel, D. Marc Kilgour, Solving the Private
Risk Problem in Brownfield Redevelopment using Fuzzy Real Options,
Proceedings of GDN 2009: International Conference on Group Decision and Negotiation Toronto, Canada, pp. 178-180. 2009
This proposed approach is based on Carlsson and Fullér’s article on the fuzzy real options [A8]. But since the private risk
is usually reflected as the volatility (σ), the transformation
method developed by Hanss [5] is integrated in order to be
able to make σ a fuzzy variable. (page 179)
A8-c20 Zhengbiao Qin, Yun Pu, and Minjie Hu, Investment Decisions in
the BOT Transport Infrastructure Applying Fuzzy Real Option, in: Proceedings of the 8th International Conference of Chinese Logistics and
Transportation Professionals - Logistics: The Emerging Frontiers of
Transportation and Development in China, pp. 1584-1589. 2008
http://dx.doi.org/10.1061/40996(330)231
A8-c19 D. Allenotor, R.K. Thulasiram, Grid resources pricing: A novel
financial option based quality of service-profit quasi-static equilibrium
model, 9th IEEE/ACM International Conference on Grid Computing,
pp. 75-84. 2008
http://dx.doi.org/10.1109/GRID.2008.4662785
Carlsson and Fullér in [A8] apply a hybrid approach to valuing real options. Their method incorporates real option, fuzzy
logic, and probability to account for the uncertainty involved
25
in the valuation of future cash flow estimates. The results
of the research efforts given in [A8] and [18] have no formal reference to the QoS that characterize a decision system.
Carlsson and Fullér [A8] apply fuzzy methods to measure
the level of decision uncertainties and did not price grid resources. (pages 76-77)
A8-c18 A.C. Tolga, C. Kahraman, Fuzzy multi-criteria evaluation of R&D
projects and a fuzzy trinomial lattice approach for real options, in: Proceedings of the 3rd International Conference on Intelligent System and
Knowledge Engineering, ISKE 2008, November 17-19, 2008, Xiamen,
China, Article number 4730966, pp. 418-423. 2008
http://dx.doi.org/10.1109/ISKE.2008.4730966
Real options give a right but not an obligation to make or
not to make an investment for a certain period. For instance,
organizing a Research and Development (R&D) laboratory
investment gives the company right to research and develop
new products now but not the obligation at future. Real options were first introduced by Trigeorgis [2]. Huchzermeier
and Loch’s [3] model built real option for R&D managers as
to when it is and when it is not worthwhile to delay commitments. The lack of past data and vague information drives
us to use fuzzy models. In the literature, fuzzy real options
valuation (FROV) models have been developed under incomplete information. First, Carlsson and Fullér [A8] developed
a heuristic real option valuation process in a fuzzy setting.
In their study present values of expected costs and expected
cash flows are calculated by trapezoidal fuzzy numbers. (page
418)
A8-c17 P. Majlender, Soft decision support systems for evaluating real and
financial investments, in: Fuzzy Engineering Economics with Applications, 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 applications, in: Fuzzy Engineering Economics with Applications, Studies in Fuzziness and Soft Computing, vol. 233/2008, Springer, [ISBN
978-3-540-70809-4], pp. 43-69. 2008
26
http://dx.doi.org/10.1007/978-3-540-70810-0_2
A8-c15 A. Dimitrovski, M. Matos, Fuzzy present worth analysis with correlated and uncorrelated cash flows, in: Studies in Fuzziness and Soft
Computing, 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 Resources
Pricing A Fuzzy Real Option Model In: IEEE International Conference
on e-Science and Grid Computing, December 10-13, 2007, Bangalore,
India, pp. 388-395. 2007
http://dx.doi.org/10.1109/E-SCIENCE.2007.37
Current literature on real option approaches to valuing projects
presents real options framework in eight categories [15]: option to defer, time-to-build option, option to alter, option to
expand, option to abandon, option to switch, growth options,
and multiple integrating options. There are also efforts reported towards improving the selection and decision methods used in the prediction of the capital that an investment
may consume. Carlsson and Fullér in [A8] apply a hybrid
approach to valuing real options. Their method incorporates
real option and fuzzy logic and some aspects of probability
to account for the uncertainty involved in the valuation of future cash ow estimates. The results of the research efforts
given in [15] and [8] have no formal reference to the QoS that
characterize a decision system. Carlsson and Fullér [8] apply
fuzzy methods to measure the level of decision uncertainties,
however, there is a lack of indication on how accurate the decisions could be. (page 389)
A8-c13 Cheng, Jao-Hong Lee, Chen-Yu , Product Outsourcing under Uncertainty: an Application of Fuzzy Real Option Approach, IEEE International Fuzzy Systems Conference, 2007 (FUZZ-IEEE 2007), 23-26
July 2007, London, [ISBN: 1-4244-1210-2], pp. 1-6. 2007
http://dx.doi.org/10.1109/FUZZY.2007.4295675
Carlsson and Fullér [A8] stated that the relevance of historic
data diminished very quickly after 2-3 years, so it was not
worthwhile to claim the predictive value of time series after 5
years (and even much more so for 15-25 years ahead). (page
2)
27
A8-c12 Chen, Tao; Zeng, Yurong; Wang, Lin; Zhang, Jinlong, Evaluating
IT Investment Using a Hybrid Approach of Fuzzy Risk Analysis and
Real Options, Fourth International Conference on Fuzzy Systems and
Knowledge Discovery (FSKD 2007), Haikou, Hainan, China, 24-27
Aug. 2007, vol.1, pp.135-139. 2007
http://dx.doi.org/10.1109/FSKD.2007.276
A8-c11 David Allenotor and Ruppa K. Thulasiram, A Grid Resources Valuation Model Using Fuzzy Real Option, in: Ivan Stojmenovic; Ruppa
K.Thulasiram; Laurence T.Yang; Weijia Jia; Minyi Guo; Rodrigo Fernandes de Mello eds., Parallel and Distributed Processing and Applications, 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 Fullér in [A8] apply a hybrid approach (real option, fuzzy logic, and probability theory) to value future options. The results given in [2] and [A8] have no formal reference to the QoS that characterize a decision system. Carlsson
and Fullér [A8] apply fuzzy methods to measure the level of
decision uncertainties, however, there is a lack of indication
on how accurate the decisions could be. (page 624)
A8-c10 Chen Tao, Zhang Jinlong, Yu Benhai, and Liu Shan A Fuzzy Group
Decision Approach to Real Option Valuation, in: Aijun An, Jerzy Stefanowski, Sheela Ramanna, Cory J. Butz, Witold Pedrycz, Guoyin
Wang (Eds.): Rough Sets, Fuzzy Sets, Data Mining and Granular Computing, 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, [ISBN
978-3-540-72529-9], 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 Real
Option Analysis for IT Investment in Nuclear Power Station, Y. Shi et
al. (Eds.): ICCS 2007, Part III, Lecture Notes in Computer Science,
Volume 4489/2007, Springer, [ISBN 978-3-540-72587-9], pp. 953959. 2007
28
http://dx.doi.org/10.1007/978-3-540-72588-6_152
Step 5: the real option valuation of the investment
In the last step, we can assess the real option value of the investment based on the result obtained above. For the purpose
of simplicity, we assume that only the expected payoff is uncertain and utilize the Black-Scholes pricing model. Then the
fuzzy real option value of an investment is [A8]
FROV = V N (d1 ) − Xe−rT N (d2 )
where
d1 =
ln(E(V )/X) + (r + σ 2 /2)T
√
,
σ T
√
d2 = d1 − σ T ,
Only V is a fuzzy number. E(V ) and σ represent, respectively, the possibilistic expected value and the standard deviation of fuzzy figure V . The computing result FROV is also
a fuzzy number, representing the real option value of the investment under consideration. (page 955)
A8-c8 Yoshida, Y. Option pricing theory in financial engineering from the
viewpoint of fuzzy logic, in: Kahraman, Cengiz (Ed.) Fuzzy Applications in Industrial Engineering, Series: Studies in Fuzziness and Soft
Computing , 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 combining randomness and fuzziness, in: Computational Intelligence, International Conference on Intelligent Computing, ICIC 2006, Kunming,
China, August 16-19, 2006. Proceedings, Part II, LECTURE NOTES
IN ARTIFICIAL INTELLIGENCE, vol. 4114, pp. 798-808. 2006
http://dx.doi.org/10.1007/11816171_100
Carlsson and Fullér [A8] gianed the present value of expected
cash flow could not usually be characterized by a single number. However, their experiences with the practical projects
showed that managers were able to estimate the present value
of expected cash flows by using a trapezoidal fuzzy variable.
(page 799)
29
A8-c6 Karsak EE, A generalized fuzzy optimization framework for R &
D project selection using real options valuation, in: Computational
Science and Its Applications - ICCSA 2006, LECTURE NOTES IN
COMPUTER SCIENCE 3982: 918-927 2006
http://dx.doi.org/10.1007/11751595_96
Lately, options valuation approach has been proposed as a
more suitable alternative for determining the benefits from
R&D projects [9]. Options approach deviates from the conventional DCF approach in that it views future investment opportunities as rights without obligations to take some action
in the future [3]. The asymmetry in the options expands the
NPV to include a premium beyond the static NPV calculation,
and thus presumably increase the total value of the project and
the probability of justification. Carlsson and Fullér [A8] further extended the use of options approach in R&D project valuation by considering the possibilistic mean and variance of
fuzzy cash flow estimates. This paper presents a novel fuzzy
formulation for R&D project selection accounting for project
interactions with the objective of maximizing the net benefit based on expanded net present value which incorporates
the real options inherent in R&D projects in a fuzzy setting.
Although a fuzzy optimization model is provided for R&D
portfolio selection in [14], the project interactions are completely ignored. Compared with the real options valuation
procedures delineated in [A8, 14], the valuation approach utilized in this paper models exercise price as a stochastic variable enabling to deal with technological uncertainties in real
options analysis, and considers both the benefits of keeping
the development option alive and the opportunity cost of delaying development. (page 919)
A8-c5 T. Sato, S. Takahashi, C, Huang and H. Inoue, Option pricing with
fuzzy barrier conditions, in: Y. Liu, G. Chen and M. Ying eds., Proceedings of the Eleventh International Fuzzy systems Association World
Congress, July 28-31, 2005, Beijing, China, 2005 Tsinghua University
Press 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 reengineering project, Soft Computing with Industrial Applications - Proceedings 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, applications and software, Advances in Fuzzy Systems - Applications and
Theory, Vol. 22, World Scientific. 2004
in Ph.D. disserations
• Markku Heikkilä, R&D investment decisions with real options - Profitability and Decision Support, Åbo Akademi University, Åbo, Finland,
[ISBN 978-952-12-2379-2]. 2009
• Péter Majlender, A Normative Approach to Possibility Theory and Soft
Decision Support, Turku Centre for Computer Science, Institute for
Advanced Management Systems Research, Åbo Akademi University,
No 54, [ISBN 952-12-1409-0]. 2004
31