A fuzzy approach for R&D compound option valuation

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ScienceDirect Fuzzy Sets and Systems 310 (2017) 108–121 www.elsevier.com/locate/fss

A fuzzy approach for R&D compound option valuation Marta Biancardi a , Giovanni Villani b,∗ a Department of Economics, University of Foggia, Largo Papa Giovanni Paolo II, 1, 71121 Foggia, Italy b Department of Economics and Mathematics, University of Bari, Via Camillo Rosalba, 53, 70124 Bari, Italy

Received 15 June 2015; received in revised form 18 October 2016; accepted 19 October 2016 Available online 25 October 2016

Abstract This paper is devoted to propose a random fuzzy methodology in order to value R&D investments combining the stochastic approach with the fuzzy analysis. As it is commonly known, an R&D project is characterized by a sequential phase in which each phase gives to the manager the opportunity to realize or not the investment. So, in real option world, this opportunity can be analyzed as a compound American exchange option (CAEO). In this paper, the fuzzy approach is used to model two important parameters of R&D evaluation: the volatility of asset V and the opportunity costs of deferring project δv . Therefore, we present a γ -level of fuzzy prices of CAEO and the fuzzy mean value of CAEO using a pessimistic–optimistic weight. © 2016 Elsevier B.V. All rights reserved. Keywords: Fuzzy analysis; Real option; R&D project valuation

1. Introduction The research of new evaluation tools for the R&D investment projects has involved many academic scholars since the traditional methodologies, such as the net present value (NPV) or the internal rate return (IIR), could mislead the manager. In fact, these projects present a negative NPV owing to low initial cash flows and high uncertainty that influences the discount rate. Therefore, the real option approach becomes very important in order to include both the managerial flexibility to grow, delay, scale down or abandon projects and to support management to take adapted decisions. As it is stated in [5], a real option is “the right, but not the obligation, to take an action (e.g. deferring, expanding, contracting, or abandoning) at a predetermined cost called the exercise price, for a predetermined period of time – the life of option”. But, especially for the evaluation of R&D investments, the exercise costs are uncertain and so the exchange options are more suitable to model the R&D projects. Moreover, an R&D project is characterized by a sequential phase in which each phase gives to the manager the opportunity to realize or not the investment. In add, we consider also the managerial flexibility to realize the development investment in the last phase and before the maturity, in order to capture the project cash flows. So, in real option world, this opportunity can be view as a * Corresponding author.

E-mail addresses: [email protected] (M. Biancardi), [email protected] (G. Villani). http://dx.doi.org/10.1016/j.fss.2016.10.013 0165-0114/© 2016 Elsevier B.V. All rights reserved.

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compound American exchange option (CAEO) evaluated following Lee and Paxson [12] model. In particular, this option is composed by a compound European exchange option (CEEO), which values is determined by Carr [3], and an early exercise premium given by the difference between the simple American exchange option (SAEO) and the simple European exchange option (SEEO). Margrabe [13] proposed a model to value a SEEO that can be exercised only at maturity. This model can value the American one only if the underlying asset does not distribute dividends, since an American option should never be exercised prior to maturity in the absence of dividends. McDonald and Siegel [14] value an European exchange option considering that the assets distribute dividends. In order to value the SAEO we use the Lee and Paxson [11] approximation; it is determined by the pseudo American exchange option (PAEO) given by Carr [3] that can be exercised in two points and by the perpetual American exchange option. In our paper, we follow the classical assumption for real option modeling assuming to have two assets that follow a geometric brownian motion. In spite of we are aware that the pure stochastic assumption is not quite suitable for R&D project evaluation since some key variables such as the future market demand, the evolution competition and the technological changes are not analyzed separately, we believe that the stochastic approach combined with fuzzy methodology gives the manager a probabilistic interval in which the R&D evaluation belongs (see [18]). As it is pointed out for instance by Carlsson and Fuller [1], Carlsson et al. [2] and Collan et al. [4], the R&D projects do not have stochastic future if proper planning has not been carried out and so the fuzzy approach requires to collect data in order to estimate future marked demand, future competition and future technology changes and so to reduce uncertainty. But, following Dubois and Prade [7], we can interpret the degree of membership μF (u) of an element u in a fuzzy set F as a degree of possibility that a parameter x has value u and consequently we can view it as degree of uncertainty. Summarizing, our contribution is collocated in the classical vision of random fuzzy analysis given by Puri and Ralescu [15], in which we extend the random outcomes of stochastic models that do not observe properly the data, with the fuzzy approach providing a special range of R&D evaluation. The main option valuations within fuzzy approach are witnessed by Carlsson and Fuller [1], Carlsson et al. [2], Yoshida et al. [17], Guerra et al. [10], Wang et al. [16] and so on. In particular, Carlsson and Fuller [1] introduced a heuristic real option rule in a fuzzy setting, where the present values of expected cash flows and expected costs were estimated by trapezoidal fuzzy numbers. Carlsson et al. [2] developed a methodology for valuing options on R&D projects, in which future cash flows were estimated by trapezoidal fuzzy number. Yoshida et al. [17] proposed the pessimistic–optimistic index in order to determine the mean value of fuzzy distribution. Guerra et al. [10] presented in their work a sensitivity analysis in fuzzy contest based on the Black–Scholes formula. Wang et al. [16] proposed the fuzzy price of compound option by fuzzifying the interest and the volatility in [9] compound option pricing formula. Starting from this last work, the main contribution of our paper is to value the CAEO handling in fuzzy manner two important parameters: the volatility σv and the opportunity costs of deferring project δv . The main differences of our paper from [16] are the analysis of R&D investment with exchange options and the evaluation of index λ for the mean fuzzy value. The paper is organized as follows. In Section 2 we introduce the stochastic models that are essential to price the CAEO and, in Section 3, we present the γ -level and the fuzzy price of CAEO. In Section 4 we develop a numerical analysis in order to enhance our results. Finally, Section 5 concludes. 2. R&D compound exchange option pricing under stochastic approach An R&D investment is characterized by a sequential phase in which in each phase the manager decides if to realize or not the investment. In our paper we assume a two-stage R&D and we assume that the Research investment R is realized at time t0 = 0, the Information Technology cost I T is spent at time t1 and the Development investment D may be made at any time between t1 and T in order to benefit of the gross project value V . Summing up, investing R at time t0 , the investor receives a first market opportunity to realize I T at time t1 , that allows to benefit from a second opportunity to make the investment D between t1 and T obtaining the gross project value V . So the second opportunity at time t1 can be valued as a simple American exchange (SAEO) option with maturity τ = T − t1 and payoff equal to max[0, V − D]. This option comprises the chance that the Development investment D will be realized if the gross project V is larger than D, otherwise manager prefers to abandon it. Coming back at time t0 , the first market opportunity gives to possibility to exchange the asset I T at time t1 and to receive the SAEO. Therefore, this opportunity is a compound American exchange option (CAEO) with maturity t1 and payoff is max[0, SAEO − I T ], as it is illustrated in Fig. 1.

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Fig. 1. R&D investment as compound American exchange option (CAEO).

As the R&D investment is characterized by high risks of failure, low expected cash flows in the initial phase but high cash flows in the Development phase, we take into account the managerial flexibility to realize D at any time between t1 and T . Therefore the compound American exchange option (CAEO) is the suitable tool to model an R&D project, because it considers both the sequential investment and the managerial flexibility. As we have pointed out, the uncertainty is a crucial parameter that characterizes the R&D investments. Dixit [6] was one of the first to suggest to use stochastic volatility model in valuing real options. In fact, when the volatility parameter increases then the value of waiting also goes up and consequently the value of option also increases. However, as the project becomes more risky, the firm must wait longer in order to capture the full value of waiting. For instance, General Motors stopped its hybrid project in 1998 and resumed it after observing Toyota’s success in its hybrid car project. In add, the development of an innovative new drug in the pharmaceutical sector is associated with many uncertainties. In this context, the regulatory authorities are more cautious and have extended their safety requirements in response to observed risks, and so companies can delay new drug launches. On the other hand, if a firm chooses to defer the exercise of its option until better information is received (thus resolving uncertainty), it runs the risk that other firms may preempt it by exercising first. Most real-life situations present preemptive characteristics. In real market terms, dividends may represent several types of opportunity costs caused by holding the real option unexercised, namely the loss of project’s cash flows when an investment opportunity is postponed. For instance, they may be considered as “competitive dividends” that incorporate the threat of competitors entry in the market that can take away cash flows. For these reasons, we decide to fuzzifying two considerable parameters and to observe their effects on the real option valuation: the volatility of asset V , namely σv and the competitive dividend yields δv . The evaluation of our R&D Compound option is mainly suitable in this context. First of all, we begin our analysis considering a compound European exchange option (CEEO), whose underlying asset is a simple European exchange option (SEEO). We consider two nonidentical risky assets, V and D, described by the following stochastic differential equations: dV = (μv − δv )dt + σv dZtv V dD = (μd − δd )dt + σd dZtd D   dV dD cov , = ρvd σv σd dt V D

(1) (2) (3)

where μv and μd are the expected rates of return, δv and δd are the corresponding dividend yields, σv2 and σd2 are the respective variance rates, ρvd is the correlation between changes in V and D with −1 ≤ ρvd ≤ +1, (Ztv )t∈[0,T ] and (Ztd )t∈[0,T ] are two Brownian processes defined on a filtered probability space (, A, F, P). Denoting by s(V , D, T − t) the value at time t of a SEEO to exchange asset D for asset V at time T , Margrabe [13] and McDonald and Siegel [14] show, under certain assumptions, that the value of a SEEO on dividend-paying assets at time t = 0 is given by: s(V , D, T ) = V e−δv T N (d1 (P , T )) − De−δd T N (d2 (P , T )) where: • P=

V ; D

(4)

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 • σ = σv2 − 2ρvd σv σd + σd2 ; • δ = δv − δd ;  2  ln a + σ2 − δ b √ • d1 (a, b) = ; d2 (a, b) = d1 (a, b) − σ b ; √ σ b • N(d) is the standard normal distribution function. Remark 1. We observe that when assets V and D are positively perfectly correlated (ρvd = +1) and simultaneously they have the same variance rates (σv = σd ), it results that σ = 0. Obviously, this supposition implies that V and D are the same and this is not provided in exchange option context, that requires two non-identical assets. Carr [3] develops a model to value the CEEO assuming that the expiration date is t1 < T , the underlying asset is a SEEO, s(V , D, τ ), whose maturity is at time τ = T − t1 , the exercise price I T = qD is stochastic and defined as a ratio q of asset D at time t1 . Carr [3] shows that the value of such CEEO is given by:     P −δv T c(s(V , D, τ ), qD, t1 ) = V e N 2 d1 , t1 , d1 (P , T ) ; ρ P∗     P , t , d , T ; ρ − De−δd T N2 d2 (P ) 1 2 P∗    P , t1 (5) − qDe−δd t1 N1 d2 P∗  t1 where ρ = and P ∗ is the critical price ratio that solves the following equation: T P ∗ e−δv τ N (d1 (P ∗ , τ )) − e−δd τ N (d2 (P ∗ , τ )) = q

(6)

P∗

In other words, is the critical price ratio that makes indifferent to exercise or not the CEEO at maturity date t1 . The CEEO will be exercised at time t1 if the price ratio P at time t1 is higher than P ∗ . Unfortunately, the SEEO considers that the delivery asset D can be realized at time T . When the dividends are considerable, it may be preferable to spend the asset D earlier in order to benefit of cash flows. Denoting by S the value of a simple American exchange option (SAEO), Lee and Paxson [11] provide an analytic approximation of SAEO at time t = 0: S(V , D, T ) = S∞ −

(S∞ − S2 )2 S∞ − s

(7)

where s is the value of SEEO, S2 is the price of a pseudo American exchange option (PAEO) that can be exercised either at maturity T and at midpoint T2 . With the American option, we cover the managerial flexibility that the investment D can be realized at any time before the maturity T in order to capture the cash flows. Following Carr [3], the value of PAEO (S2 ) is:        T P T P T 2N S2 (V , D, T ) = V e−δv T N2 −d1 , (P , T ), −ρ d , , d + V e 1 1 1 P ∗ 2 P ∗ 2        T P T P T −δd T 2 − De N2 −d2 , , , d2 (P , T ), −ρ1 − De N d2 (8) P ∗ 2 P ∗ 2   with ρ1 = TT/2 = 12 . Let us define P ∗ the critical price ratio that makes indifferent the option exercise or not at mid-life time

T 2.

T P ∗ e−δv 2

It solves the following equation: 

     ∗ T −δd T2 ∗ T N d1 P , N d2 P , −e = P ∗ − 1 2 2

(9)

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Moreover, the value of the perpetual American exchange option S∞ is given by:   D P θ S∞ (V , D) = ∗ θ − 1 P∞

(10)

∗ denotes the optimal exercise price at which the perpetual American exchange option should be exercised: where P∞ ∗ P∞ =

θ θ −1

(11)

and: θ=

(δv +

σ2 2 )+

 (δv +

σ2

σ2 2 2 )

(12)

.

Remark 2. We can observe that lim θ = +∞, σ →0

lim θ = 1 and the first derivative

σ →+∞

∂θ ∂σ

< 0. So it is possible to

affirm from Eqs. (11) and (12) that θ > 1. About the first derivative of S∞ with respect to θ we compute that 

D

P (θ−1) θ

θ

  ln P (θ−1) θ

∂S∞ ∂θ

=

. It is easy to prove that, when P < 1 then ∂S∂θ∞ < 0 for each θ value otherwise, when P > 1 we  P . This consideration will be very helpful within the fuzzy analysis of perpetual have that ∂S∂θ∞ < 0 for θ ∈ 1, P −1 American exchange option. θ−1

Let denote by C the value of a compound American exchange option (CAEO) given by Lee and Paxson [12]. This option is the sum of the CEEO with maturity t1 and the early exercise premium given by the difference between the SAEO that can be exercised at any time during t1 and T and the SEEO with maturity τ = T − t1 , i.e.: C(S(V , D, τ ), qD, t1 ) = c(s(V , D, τ ), qD, t1 ) + S(V , D, τ ) − s(V , D, τ ) where

(13)

P ∗

solve the following equation: τ s P ∗ , 1, (14) = P ∗ − 1 2 Finally, investing R at time t0 , manager obtains the investment opportunity which value is given by CAEO. Consequently, the Research investment R will be realized at time t0 if the profit given by: 

= −R + C(S(V , D, τ ), qD, t1 )

(15)

is positive. Moreover, this profit can be compared with others alternative performances in order to reject or not the project but, the fundamental aim of our paper, is to give a fuzzy value of CAEO. 3. Fuzzy approach to value a compound American exchange option (CAEO) Let X a nonempty set. A fuzzy set A in X is defined by a membership function μA : X → [0, 1] and μA (x) is interpreted as the degree of membership of element x in fuzzy set A for each x ∈ X. When the function μA (x) = 0 means that the element x ∈ X is not a membership of A and, on the other hand, while μA (x) = 1 we affirm the absolute membership of x in A. The other values between zero and one denote the degree of membership of x in A. In particular, a fuzzy subset A of X is called normal if there exists an element x ∈ X such that A(x) = 1. Moreover, using the theory of fuzzy subsets, we can define a fuzzy number A as a fuzzy set of real line with a normal, convex and continuous membership function of bounded support. Definition 1. The fuzzy set A is called triangular fuzzy number with core ac , left width α and right width β if its membership function is defined by: ⎧ a −x if ac − α ≤ x ≤ ac ; ⎨ 1 − cα c if ac ≤ x ≤ ac + β; μA (x) = 1 − x−a (16) β ⎩ 0 otherwise A is completely determined by the function μA and their use is interchangeable.

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In other words, a triangular fuzzy number can be denoted as A = (ac − α, ac , ac + β) with core ac and the support of A is (ac − α, ac + β). If A(x) = 1 then x belongs to A with a degree equal to one, and if A(x) = 0 then x belongs to A with a degree equal to zero (x ∈ / (ac − α, ac + β)), i.e. x is too far from ac . In the other cases, if 0 < A(x) < 1 then x is close to ac . Let denote by: [A]γ = [ac − (1 − γ )α, ac + (1 − γ )β] [A]γ

∀ γ ∈ [0, 1]

(17) [A]γ

the γ -level of A. If γ = 1 then = ac while if γ = 0 then = A. A fuzzy set A is called convex fuzzy set if [A]γ is convex ∀ γ ∈ [0, 1], i.e. μA (γ x + (1 − γ )y) ≥ min{μA (x), μA (y)}, ∀γ ∈ [0, 1]. In other words, if A is a fuzzy number, then the γ -level of A is a closed interval and we can denote [A]γ = [aγL , aγR ] with aγL = min[A]γ and aγR = max[A]γ . In order to determine the left and the right width of a fuzzy set, we use the extension principle proposed in Definition 2. Definition 2. Let A1 , A2 , · · · , An fuzzy sets defined on X1 , X2 , · · · , Xn and B = f (A1 , A2 , · · · , An ). The membership function of B is defined as: μB (y) =

max

y=f (x1 ,x2 ,··· ,xn )

{min[μA1 (x1 ), μA2 (x2 ), · · · , μAn (xn )]}

Applying the extension principle to arithmetic operations, it is possible to define fuzzy arithmetic operations. Let U , Z, Y denote the fuzzy sets that represent the operands u, z and y, respectively. Using the extension principle we have that: μY (y) = max {min[μU (u), μZ (z)]} y=u z

R L R where denotes the fuzzy arithmetic operation. In particular, if U = [uL γ , uγ ] and Z = [zγ , zγ ] are two fuzzy interval with lower and upper bounds, then U ⊕ Z, U Z, U ⊗ Z, U  Z are also fuzzy numbers and their γ -level sets are: 

L R R (U ⊕ Z)γ = uL γ + zγ , uγ + zγ ; 

R R L (U Z)γ = uL γ − zγ , uγ − zγ ;   L L L R R L R R 

 L L R R L R R (U ⊗ Z)γ = min uL γ zγ , uγ zγ , uγ zγ , uγ zγ , max uγ zγ , uγ zγ , uγ zγ , uγ zγ ;      L R R L R R uL uL γ uγ uγ uγ γ uγ uγ uγ (18) (U  Z)γ = min L , R , L , R , max L , R , L , R zγ zγ zγ zγ zγ zγ zγ zγ

with the assumption that zγH = 0, for H = L, R. With the last assumption, we take into account that the element zero does not belong to the support of Z and so the division operation on fuzzy numbers is admissible. Through the fuzzy approach, we consider two important parameters as fuzzy numbers: the volatility σ˜v of asset V and dividend yields δ˜v . This choice depends on the consideration that R&D projects are characterized by high uncertainty in their results and so the volatility σv is a fundamental parameter that reflects the R&D evaluation. Moreover, the dividends δv represent the opportunity costs of deferring the project implying the loss of project cash flows. But, in real market contest, dividends may represent other opportunity costs holding the real options unexercised. Therefore, the fuzzy analysis allows us to determine a γ -level set, denoted as [C]γ = [CγL , CγR ], in which the CAEO price belongs with a membership degree between zero and one. As illustrated by Zadeh [18] and Dubois and Prade [8], the membership function of fuzzy sets can be interpreted as a possibility distribution, namely a mapping π from S to a totally order scale L, with top 1 and bottom 0. This function represents the state of knowledge of an agent distinguishing what is acceptable and what is less acceptable, with the convention that π(s) = 0 means that the state s is rejected while π(s) = 1 means that s is totally plausible. Then the interval [C]γ = [CγL , CγR ] is the interval containing the values of all random variables of C whose core is the value CS given by Eq. (13). In this fashion, as in the real environment it is very hard to make careful forecasts, within the fuzzy numbers we consider also the human inaccuracy of prediction determining a γ -interval in which the R&D evaluation belongs. Consequently, a possibility distribution of R&D evaluation on a set [C]γ can represent for the management both a preference of distribution that models a plausibility profile describing the degree of R&D project acceptance and also as a degree of uncertainty about R&D evaluation.

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Starting from Eq. (13), first of all we analyze the pricing of CEEO c(s, qD, t1 ). Let denote by (δv )L γ := δv − (1 − L R γ )αδ and (δv )γ := δv + (1 − γ )βδ the left-end point and the right-end point of γ -level of δv ; by (σv )γ := σv − (1 − γ )ασ and (σv )R γ := σv + (1 − γ )βσ the left-end point and the right-end point of γ -level of σv . Obviously, we carry out L our analysis with the assumption that the left fuzzy points (δv )L γ and (σv )γ are still strictly positive quantities in order to have a positive support, and so the choice of αδ , ασ and γ are made in this direction. Finally we denote by δγL :=   R := (δ )R − δ ; σ L := ((σ )L )2 + σ 2 − 2ρ σ (σ )L and σ R := ((σ )R )2 + σ 2 − 2ρ σ (σ )R . (δv )L − δ ; δ d v d v vd d v v γ vd d v γ γ γ γ γ γ γ γ d d Using the formulas illustrated in the extension principle of Definition 2, we can write the γ -set of CEEO as [c]γ = [cγL , cγR ] that can be computed as:       L R R P P −δd T cγL = V e−(δv )γ T N2 d1 , t1 , d1 (P , T )L N 2 d2 , t1 , d2 (P , T )R γ ; ρ − De γ ;ρ P∗ P∗ γ γ   R  P −δd t1 − qDe N 1 d2 , t1 ; P∗ γ       R L LT P P −(δ ) −δd T , t1 , d1 (P , T )R N 2 d2 , t1 , d2 (P , T )L cγR = V e v γ N2 d1 γ ; ρ − De γ ;ρ P∗ P∗ γ γ   L  P −δd t1 − qDe N 1 d2 , t1 ; (19) P∗ γ where:



aγH

= ln



P (P ∗ )

e H

−δγH t1

+

γ

  H dγH = ln P e−δγ T +



2

(σγH )

2 (σγH )

2 T;

t1 ;

= ln

bγH

P (P ∗ )

 e H

−δγH t1

2

−

γ

  H fγH = ln P e−δγ T −

2 (σγH )

T;

(σγH ) 2

√ gγH = σγH t1 ;

t1 ;

√ H T; hH γ = σγ

2 2 for H = L, R. Moreover, using Definition 2, we obtain:      L  R aγL aγL aγR aγR aγL aγL aγR aγR P P d1 , t1 = min , , , , t1 = max , , , ; d1 ; P∗ gγL gγR gγL gγR P∗ gγL gγR gγL gγR γ γ 

d1 (P , T )L γ  d2

P , t1 P∗

dγL dγL dγR dγR = min , R, L, R hL γ hγ hγ hγ L γ



 ;



bγL bγL bγR bγR = min , , , gγL gγR gγL gγR 

fγL fγL fγR fγR d2 (P , T )L , R, L, R γ = min hL γ hγ hγ hγ

d1 (P , T )R γ 

 ;

d2

dγL dγL dγR dγR = max , R, L, R hL γ hγ hγ hγ

P , t1 P∗

R γ

;

;



bγL bγL bγR bγR = max , , , gγL gγR gγL gγR 





fγL fγL fγR fγR , R, L, R d2 (P , T )R γ = max hL γ hγ hγ hγ

 ;

 .

Proof 1. The γ -set of [c]γ = U (Z ⊕ Y ) where: R U = [uL γ , uγ ] 

    L R P P −(δv )L T L R γ N 2 d1 , t1 , d1 (P , T )γ ; ρ , V e N 2 d1 , t1 , d1 (P , T )γ ; ρ ; = Ve P∗ P∗ γ γ        L R P P L R −δd T L −δd T R Z = [zγ , zγ ] = De N 2 d2 , t1 , d2 (P , T )γ ; ρ , De N 2 d2 , t1 , d2 (P , T )γ ; ρ ; P∗ P∗ γ γ −(δv )R γT





M. Biancardi, G. Villani / Fuzzy Sets and Systems 310 (2017) 108–121

 Y

= [yγL , yγR ] =

qDe

 −δd t1

N 1 d2



P , t1 P∗

L 

 , qDe

−δd t1

N 1 d2

γ



P , t1 P∗

115

R  ; γ

because the bivariate standard normal distribution N2 (a, b, ρ) and the standard normal distribution N1 (a) are increasing while the function e−x is decreasing. 2 Remark 3. The fuzzy values gγH and hH different from zero. γ with H = L, R, are strictly positive quantities and so √ √ First of all, the time t1 and T are strictly greater than zero and so this assures us that t1 > 0 and T > 0. In add, the left-fuzzy value is (σv )L γ > 0 if and only if: 2 2 ((σv )L 2  γ ) + σd σγL > 0 ⇔ (σv )L + σd2 − 2ρvd σd (σv )L γ γ > 0 ⇔ ρvd < 2σd (σv )L γ

Obviously we observe that 2 2 ((σv )L γ ) + σd

2σd (σv )L γ

>1⇔



 2 (σv )L γ − σd > 0

Since ρvd ∈ [−1, 1], we have that condition ρvd <

2 2 ((σv )L γ ) + σd

is always verified and so σγL > 0 and consequently 2σd (σv )L γ gγL and hL are strictly fuzzy positive values. The unique case for which σγL = 0 is when simultaneously (σv )L γ = σd γ and ρvd = +1 but the financial meaning of this assumption is that assets V and D are the same and this is not possible in exchange option modeling (see Remark 1). Finally, with the same approach we prove that gγR > 0 and hR γ > 0. Recalling Eq. (6), (P ∗ )H γ is the unique term that solves the following equation: ⎛  ⎛  ⎞   H 2 H ∗ )H e−δγH τ + (σγ ) τ ln (P ln (P ∗ )H e−δγ τ −  ∗ H −(δv )H τ ⎜ γ γ 2 ⎟ −δ τ ⎜ γ N P γe √ √ ⎝ ⎠−e d N⎝ σγH τ σγH τ

(σγH )2 τ 2

⎞ ⎟ ⎠=q

for H = L, R. Now, using the same methodology, we value the SEEO within fuzzy environment with maturity τ . The γ -level of SEEO is denoted by [s]γ = [sγL , sγR ] and it is equal to:  L   R  R sγL = V e−(δv )γ τ N d1 (P , τ ) γ − De−δd τ N d2 (P , τ ) γ  R   L  L sγR = V e−(δv )γ τ N d1 (P , τ ) γ − De−δd τ N d2 (P , τ ) γ (20) where:    (σ H )2  (σ H )2 √ H H γ γ jγH = ln P e−δγ τ + τ ; kγH = ln P e−δγ τ − τ ; yγH = σγH τ 2 2     L jR jL jR L jR jL jR j j γ γ γ γ γ γ γ γ (d1 (P , τ ))L , , , , , , ; (d1 (P , τ ))R γ ) = min γ ) = max yγL yγL yγR yγR yγL yγL yγR yγR     kγL kγR kγL kγR kγL kγR kγL kγR L R (d2 (P , τ ))γ ) = min , , , , , , ; (d2 (P , τ ))γ ) = max yγL yγL yγR yγR yγL yγL yγR yγR Proof 2. By Definition 2, the γ -set of [s]γ = U Z where:  −(δv )R τ 

L   R  L R γ N U = uL d1 (P , τ ) γ , V e−(δv )γ τ N d1 (P , τ ) γ ; γ , uγ = V e   L   R 

Z = zγL , zγR = De−δd τ N d2 (P , τ ) γ , De−δd τ N d2 (P , τ ) γ because the standard normal distribution N (a) is increasing while e−x is a decreasing function.

2

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After that, using the formula illustrated in Eq. (7), we present the γ -level of perpetual American exchange option R [S∞ ]γ = [(S∞ )L γ , (S∞ )γ ] that can be computed as: (S∞ )L γ

D = L θγ − 1

D (S∞ )R γ = R θγ − 1 ∗ )L = where (P∞ γ

θγL θγL −1





P ∗ )L (P∞ γ P

θγL θγR (21)

∗ )R (P∞ γ

∗ )R = and (P∞ γ

θγR . θγR −1

Proof 3. We assign as:   H 2    2 2 H H + −(δv )H mH = − −(δ ) − 0.5 σγ ; v γ γ γ − 0.5 σγ

 L 2 nH ; γ = σγ

for H = L, R. As analyzed in Remark 2, the value of S∞ is decreasing with respect to θ when P < 1 and with respect to θ ∈ ]1, PP−1 [, when P > 1. In case of decreasing of S∞ we assign as:     R L R R L R mL mL γ mγ mγ mγ γ mγ mγ mγ L R θγ = max , L , R , R ; θγ = min , L, R, R ; nL nγ nγ nγ nL nγ nγ nγ γ γ otherwise we assign:   R L R mL γ mγ mγ mγ L θγ = min , L, R, R ; nL nγ nγ nγ γ R so that (S∞ )L γ < (S∞ )γ .



θγR

R L R mL γ mγ mγ mγ = max , , , R nR nL nL γ γ nγ γ



2

In add, we determine the value of PAEO using the fuzzy approach. Also in this case, the γ -level of PAEO with R maturity τ , i.e. S2 (V , D, τ ) is [S2 ]γ = [(S2 )L γ , (S2 )γ ] and, using the rules given in Definition 2, it can be computed as:         τ P τ L P τ L −(δv )R τ L L γ (S2 )γ = V e N2 −d1 , , d1 (P , τ )γ , −ρ1 + V e 2 N d1 , P ∗ 2 γ P ∗ 2 γ         τ P τ R P τ R −δd τ R − De N2 −d2 , , d2 (P , τ )γ , −ρ1 − De 2 N d2 , P ∗ 2 γ P ∗ 2 γ         τ P τ R P τ R −(δv )L τ R R γ (S2 )γ = V e N2 −d1 , , d1 (P , τ )γ , −ρ1 + V e 2 N d1 , P ∗ 2 γ P ∗ 2 γ     L L   τ τ τ P P 2 − De−δd τ N2 −d2 , , d2 (P , τ )L , (22) γ , −ρ1 − De N d2 P ∗ 2 γ P ∗ 2 γ where: pγH

 = ln

P

 e H

(P ∗ )γ

−δγH τ



2

+

(σγH ) 2

τ;

qγH

for H = L, R. So we obtain that:     pγL pγR pγL pγR P τ L d1 , = min , , , ; P ∗ 2 γ rγL rγL rγR rγR

= ln

 d1

P

 e H

(P ∗ )γ

P τ , P ∗ 2

−δγH τ

R γ

−

(σγH )



2

2

τ;

√ rγH = σγH τ ;

pγL pγR pγL pγR = max , , , rγL rγL rγR rγR

 ;

M. Biancardi, G. Villani / Fuzzy Sets and Systems 310 (2017) 108–121

 d2

P τ , P ∗ 2

L γ



qγL qγR qγL qγR = min L , L , R , R rγ rγ rγ rγ



 ;

d2

P τ , P ∗ 2

R γ



qγL qγR qγL qγR = max L , L , R , R rγ rγ rγ rγ

117

 ;

R L R and d1 (P , τ )L γ ; d1 (P , τ )γ ; d2 (P , τ )γ ; d2 (P , τ )γ have been defined previously in the evaluation of SEEO. Moreover, ∗ H (P )γ , for H = L, R is the critical value that solves the following equation: ⎛  ⎛   (σ H )2 τ ⎞  (σ H )2 τ ⎞ −δγH τ2 −δγH τ2 γ 2 ∗ ∗ H H ln (P )γ e + 2 − γ2 2  ∗ H −(δv )H τ ⎜ ln (P )γ e τ ⎜ ⎟ ⎟ −δ d γ 2N   P γ e ⎝ ⎠−e 2N⎝ ⎠ τ τ H H σγ 2 σγ 2  H = P ∗ γ − 1

Proof 4. The γ -set of [S2 ]γ = (U ⊕ X) (Z ⊕ Y ) where: 

R U = uL γ , uγ          P τ L P τ R −(δv )R τ −(δv )L τ L R γ γ N2 −d1 , , d1 (P , τ )γ , −ρ1 , V e N2 −d1 , , d1 (P , τ )γ , −ρ1 ; = Ve P ∗ 2 γ P ∗ 2 γ         

L R τ τ P τ L P τ R X = xγ , xγ = V e 2 N d1 , , , V e 2 N d1 ; P ∗ 2 γ P ∗ 2 γ 

Z = zγL , zγR          P τ L P τ R −δd τ L −δd τ R N2 −d2 , , d2 (P , τ )γ , −ρ1 , De N2 −d2 , , d2 (P , τ )γ , −ρ1 ; = De P ∗ 2 γ P ∗ 2 γ         

L R τ τ P τ L P τ R 2 2 Y = yγ , yγ = De N d2 , , , De N d2 P ∗ 2 γ P ∗ 2 γ because the bivariate standard normal distribution N2 (a, b, ρ) and the standard normal distribution N1 (a) are increasing while the function e−x is decreasing. 2 Finally, using the addition principle of fuzzy numbers, we have that the γ -level of CAEO is equal to [C]γ = where CγL = cγL + SγL − sγL and CγR = cγR + SγR − sγR . Let denote by CF the fuzzy value of CAEO. This value is given by the mean of [C]γ using as weight the pessimistic–optimistic index λ defined by Yoshida et al. [17] AR , as it is shown in Fig. 2, namely: where λ = AL+AR

[CγL , CγR ],

1 E(C) =

! (1 − λ)CγL + λCγR d γ

(23)

0

So, fuzzy value of CAEO denote by CF is CF =

(1 − λ)CγL + CS + λCγR

(24) 2 where CS is the stochastic value of CAEO given by Eq. (13). Therefore, we obtain a distribution around the of stochastic CAEO. Consequently, the fuzzy profit is F = −R + CF and the project will be realized if F is greater than zero. 4. Numerical R&D applications In this section, we present same numerical applications in order to show our method. We start from project n. 1 and after that we change others parameter values to verify the effects on fuzzy CAEO analysis. First of all, Table 1 summarizes input values of several R&D projects assuming as current time t0 = 0.

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Fig. 2. Fuzzy triangular numbers of CAEO.

Table 1 Input values of R&D project investments. n

V

D

IT

t1

T

σv

σd

ρvd

δv

δd

αδ

βδ

ασ

βσ

γ

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20

100 120 80 100 100 100 100 100 100 100 100 100 100 100 100 100 100 100 100 100

120 120 120 140 80 120 120 120 120 120 120 120 120 120 120 120 120 120 120 120

40 40 40 40 40 60 20 40 40 40 40 40 40 40 40 40 40 40 40 40

1 1 1 1 1 1 1 1 1 0.75 1.50 1 1 1 1 1 1 1 1 1

2 2 2 2 2 2 2 2.5 3 2 2 2 2 2 2 2 2 2 2 2

0.80 0.80 0.80 0.80 0.80 0.80 0.80 0.80 0.80 0.80 0.80 0.95 0.70 0.80 0.80 0.80 0.80 0.80 0.80 0.80

0.30 0.30 0.30 0.30 0.30 0.30 0.30 0.30 0.30 0.30 0.30 0.30 0.30 0.45 0.15 0.30 0.30 0.30 0.30 0.30

0.25 0.25 0.25 0.25 0.25 0.25 0.25 0.25 0.25 0.25 0.25 0.25 0.25 0.25 0.25 0.25 0.25 0.25 0.25 0.25

0.20 0.20 0.20 0.20 0.20 0.20 0.20 0.20 0.20 0.20 0.20 0.20 0.20 0.20 0.20 0.10 0.20 0.20 0.20 0.20

0.05 0.05 0.05 0.05 0.05 0.05 0.05 0.05 0.05 0.05 0.05 0.05 0.05 0.05 0.05 0.05 0.05 0.05 0.05 0.05

0.05 0.05 0.05 0.05 0.05 0.05 0.05 0.05 0.05 0.05 0.05 0.05 0.05 0.05 0.05 0.05 0.05 0.05 0.05 0.05

0.06 0.06 0.06 0.06 0.06 0.06 0.06 0.06 0.06 0.06 0.06 0.06 0.06 0.06 0.06 0.06 0.06 0.06 0.06 0.06

0.08 0.08 0.08 0.08 0.08 0.08 0.08 0.08 0.08 0.08 0.08 0.08 0.08 0.08 0.08 0.08 0.08 0.08 0.08 0.08

0.07 0.07 0.07 0.07 0.07 0.07 0.07 0.07 0.07 0.07 0.07 0.07 0.07 0.07 0.07 0.07 0.07 0.07 0.07 0.07

0.50 0.50 0.50 0.50 0.50 0.50 0.50 0.50 0.50 0.50 0.50 0.50 0.50 0.50 0.50 0.50 0.50 0.70 0.85 0.95

In particular, V is the gross project value obtained discounting the R&D expected cash flows. The Development investment D is the present capital in order to receive the asset value V . The asset D takes into account also the operating expenses, the taxes, the depreciation and so on. As our model deals with a two-stages R&D plan, we consider the Research investment R spent at time t0 and the Technology investment I T held at time t1 . The latter is a ratio q of asset D; so for our computations we have that q = IDT . The investment R is not contemplated in the CAEO formula. But investing R at time t0 we obtain the R&D opportunity evaluated as CAEO. We assume that R = 6 for each project. Following the net present value (NPV) approach, for our adapted parameter values in Table 1, it results that the NPV is always negative since V − (R + D + I T ) < 0 and so the management should reject all R&D projects. But, in real option environment, the manager will realize the investment R at time t0 if R is larger than CAEO. After time T , each R&D opportunities disappears and so the investment D can be realized between t1 and T . Conveniently, we assume that the quoted shares and trade options of similar company are adequate proxies in order to value the volatility of assets V and D and their correlation ρvd . As the R&D project are riskiness, we assume that the σv is between 0.70 and 0.95. Similar to financial options, δv is the opportunity cost of deferring the project that

M. Biancardi, G. Villani / Fuzzy Sets and Systems 310 (2017) 108–121

119

Table 2 Numerical valuations of R&D project investments using stochastic models. n

c

s

S2

S∞

S

CS

P∗

∗ P

∗ P∞

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20

7.7931 12.6495 4.1014 6.5453 11.5404 5.7044 11.2551 6.9269 6.0829 5.9425 11.0804 11.9979 5.3390 8.7367 7.6575 13.2334 7.7931 7.7931 7.7931 7.7931

16.1373 24.8798 9.0676 12.7147 26.9864 16.1373 16.1373 18.2204 19.0813 17.3801 11.6272 20.7341 13.1502 17.2122 15.9801 20.5459 16.1373 16.1373 16.1373 16.1373

16.9699 26.5312 9.3889 13.2249 29.2726 16.9699 16.9699 19.8292 21.5395 18.5867 11.8576 21.7248 13.8771 18.0820 16.8072 20.8324 16.9699 16.9699 16.9699 16.9699

29.1949 39.4831 20.1767 26.3881 38.0871 29.1949 29.1949 29.1949 29.1949 29.1949 29.1949 36.5070 14.5556 30.9661 28.9328 44.8454 29.1949 29.1949 29.1949 29.1949

17.7495 27.9959 9.7009 13.7160 31.0880 17.7495 17.7495 21.2021 23.4003 19.6701 12.0851 22.6533 24.0843 18.8969 17.5816 21.1155 17.7495 17.7495 17.7495 17.7495

9.4053 15.7656 4.7347 7.5465 15.6420 7.3166 12.8673 9.9086 10.4019 8.2325 11.5383 13.9171 6.7443 10.4214 9.2590 13.8029 9.4053 9.4053 9.4053 9.4053

1.2428 1.2428 1.2428 1.1556 1.5221 1.5221 0.9105 1.2371 1.2515 1.2370 1.2784 1.1536 1.3028 1.2216 1.2459 1.1245 1.2428 1.2428 1.2428 1.2428

1.4535 1.4535 1.4535 1.4535 1.4535 1.4535 1.4535 1.4660 1.4632 1.4624 1.4033 1.5929 1.3700 1.4849 1.4490 1.7510 1.4535 1.4535 1.4535 1.4535

2.5250 2.5250 2.5250 2.5250 2.5250 2.5250 2.5250 2.5250 2.5250 2.5250 2.5250 3.1250 2.1875 2.6562 2.5062 4.0500 2.5250 2.5250 2.5250 2.5250

implies the loss of project’s cash flows. For high values of δv , the value of the option goes to zero and so the NPV is a valid method. Finally the last columns of Table 1 contain the left and right spread of parameters δv and σv used for triangular fuzzy numbers and the γ level. With the parameter values listed in Table 1, we are able to compute the option values using the stochastic models presented in Section 2 for each R&D investment. These results are represented in Table 2. The first column denotes the CEEO (c) using Carr [3] formula while the 6th column contains the evaluation of CAEO using Eq. (13) that we denote by CS . It is possible to remark that CS is larger than c because CS takes into account the managerial flexibility that the investment D can be realized between t1 and T . For instance, comparing projects n. 1 and n. 17 in which the dividend δv increases to δv = 0.30, the CAEO goes down since it is not favourable to postpone its exercise and so to keep the option alive. In add, through the comparison between project n. 1 and n. 12 in which the volatility σv increases, we can observe that CAEO CS increases too. The other columns summarize the EEO (s), the PAEO (S2 ), the perpetual American exchange option (S∞ ), the AEO (S) that are used to compute the early exercise premium of CAEO. Moreover, the last three ones contain the critical values P ∗ for CEEO, P ∗ for ∗ for perpetual American exchange option. PAEO and P∞ Finally, Tables 3 and 4 summarize the results using the fuzzy analysis. In particular, Table 3 contains the left and right γ -side of fuzzy option values that we use to compute the CAEO by triangular fuzzy numbers. We display the left and the right γ -level of CEEO cγ , EEO sγ , PAEO (S2 )γ , perpetual American exchange option (S∞ )γ , respectively. For these values we have assumed that γ = 0.50. We can observe that for projects n. 18, 19, 20, for which γ level increases, the range size that contains the fuzzy option value diminishes. In fact when γ = 1, we have that the fuzzy values converge to stochastic pricing since we do not take into account the uncertainty of two fuzzy parameters: the volatility of asset V and the dividend yield δv . It is very interesting to compare the fuzzy values of CAEO in Table 4, i.e. CF , with the stochastic ones in Table 2, i.e. CS . We can underline that the fuzzy values CF are below to stochastic values CS since the former incorporate more uncertainty than latter, and they give to the manager a fuzzy interval [CγL , CγR ] in which can be the CAEO value. Obviously this interval depends on the γ level assumed and on the left-right spreads αδ , βδ , ασ and βσ . Moreover, Table 4 contains also the pessimistic–optimistic parameter λ, the ∗ and the last column the profit = −R + C . Following this last rule, left-right fuzzy critical values P ∗ , P ∗ and P∞ F F the bold values denote projects n. 3 and 13 with have a loss and so manager should reject them. We can observe that the best project is the n. 2 with a profit F = 9.5751.

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Table 3 Numerical valuations of R&D project investments using fuzzy approach. n

cγL

cγR

sγL

sγR

(S2 )L γ

(S2 )R γ

(S∞ )L γ

(S∞ )R γ

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20

2.5404 6.1073 0.2527 1.5604 5.7984 0.5017 6.0512 1.3883 0.3388 0.8473 5.4882 6.7708 0.0124 3.5935 2.3258 6.5622 2.5404 4.6465 6.2223 7.2701

12.5569 18.7280 7.4829 10.9863 16.9413 10.3457 16.0640 11.9130 11.2261 10.5749 16.1186 16.7582 10.1433 13.4306 12.4840 19.4445 12.5569 10.6814 9.2476 8.2800

11.4077 19.2237 4.9479 7.6552 22.3683 11.4077 11.4077 12.6815 12.9223 12.2571 7.8251 15.9608 7.9387 12.7380 11.0696 15.3638 11.4077 13.2798 14.7017 15.6573

20.6592 30.4167 12.9135 17.4764 31.5703 20.6592 20.6592 23.4075 24.7640 22.2204 15.3693 25.2908 18.1736 21.4637 20.6868 25.5698 20.6592 18.8365 17.4808 16.5836

11.7551 20.5572 4.8849 7.6662 24.9254 11.7551 11.7551 13.7576 14.8171 12.9454 7.7090 16.6010 8.1257 13.2265 11.3626 15.2129 11.7551 13.8212 15.3888 16.4415

22.0068 32.4107 13.6289 18.5030 33.5942 22.0068 22.0068 25.5853 27.8223 23.9807 15.9596 26.6482 19.4616 22.7449 22.0757 26.3340 22.0068 19.9764 18.4663 17.4670

20.3557 29.4552 12.9503 17.3761 30.8658 20.3557 20.3557 20.3557 20.3557 20.3557 20.3557 26.2376 16.3734 22.3020 19.8366 28.8846 20.3557 23.4430 26.1251 28.1236

42.3521 54.3198 31.2314 40.0349 49.1077 42.3521 42.3521 42.3521 42.3521 42.3521 42.3521 52.0344 35.4296 43.5047 42.7274 77.8613 42.3521 36.2609 32.4721 30.2372

Table 4 Numerical valuations of CAEO using fuzzy approach. n

(P ∗ )L γ

(P ∗ )R γ

∗ )L (P γ

∗ )R (P γ

∗ )L (P∞ γ

∗ )R (P∞ γ

CγL

CγR

λ

CF

F

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20

1.2356 1.2356 1.2356 1.1510 1.5062 1.5062 0.9129 1.2203 1.2670 1.2253 1.2778 1.1480 1.2937 1.2125 1.2401 1.1180 1.2356 1.2385 1.2407 1.2421

1.2590 1.2590 1.2590 1.1687 1.5482 1.5482 0.9156 1.2671 1.4097 1.2598 1.2837 1.1678 1.3209 1.2392 1.2610 1.1392 1.2590 1.2525 1.2477 1.2444

1.4668 1.4668 1.4668 1.4668 1.4668 1.4668 1.4668 1.4859 1.4721 1.4792 1.4083 1.6142 1.3791 1.5037 1.4597 1.8419 1.4668 1.4610 1.4571 1.4547

1.4316 1.4316 1.4316 1.4316 1.4316 1.4316 1.4316 1.4369 1.3977 1.4367 1.3912 1.5616 1.3533 1.4582 1.4290 1.6689 1.4316 1.4397 1.4463 1.4511

1.9740 1.9740 1.9740 1.9740 1.9740 1.9740 1.9740 1.9740 1.9740 1.9740 1.9740 2.3227 1.7709 2.0823 1.9461 2.5028 1.9740 2.1489 2.3154 2.4494

3.7396 3.7396 3.7396 3.7396 3.7396 3.7396 3.7396 3.7396 3.7396 3.7396 3.7396 5.1641 3.0262 3.8790 3.7843 16.9033 3.7396 3.1020 2.7744 2.6012

3.2217 8.6005 0.1261 1.5822 10.1431 1.1830 6.7325 3.3895 3.6454 2.1654 5.2550 8.0114 0.3821 4.5455 2.9019 6.2733 3.2217 5.7003 7.5551 8.7891

15.1685 22.5497 8.8858 12.9928 20.7556 12.9573 18.6756 16.0181 16.8110 13.9414 17.2863 19.4042 12.6231 15.9185 15.1742 20.9616 15.1685 12.8867 11.1538 9.9897

0.5175 0.5136 0.5261 0.5226 0.5181 0.5209 0.5136 0.5161 0.5131 0.5152 0.5222 0.5183 0.5197 0.5166 0.5180 0.5126 0.5175 0.5155 0.5141 0.5132

9.1951 15.5751 4.5060 7.2875 15.4493 7.0702 12.7041 9.7038 10.2282 8.0534 11.2706 13.7078 6.5026 10.2320 9.0381 13.6174 9.1951 9.2935 9.3545 9.3894

3.1951 9.5751 −1.4940 1.2875 9.4493 1.0702 6.7041 3.7038 4.2282 2.0534 5.2706 7.7078 0.5026 4.2320 3.0381 7.6174 3.1951 3.2935 3.3545 3.3891

5. Concluding remarks In our paper we have presented a fuzzy approach in order to value R&D investments. As we have seen, R&D projects can be valued as compound American exchange option since, in this manner, we take into account the sequential frame of R&D investment and the managerial flexibility to realize the development investment before the maturity capturing the project cash flows. Fuzzifying two important parameters characterizing R&D, namely the volatility of asset V and the dividends δv , we have determined a γ -level of fuzzy price and the fuzzy mean of CAEO, using the pessimistic–optimistic parameter λ proposed by Yoshida et al. [17]. Finally, we have implemented a numer-

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ical analysis showing our methodology that combines the result of stochastic models with fuzzy approach setting our contribution about the R&D compound option valuation in the random fuzzy context. Moreover, the fuzzy pricing proposed in our paper undervalues the R&D evaluation with respect to stochastic model because it considers more uncertainty about the volatility σv and the dividends δv . Acknowledgements Many thanks to the anonymous referees for the helpful comments. References [1] C. Carlsson, R. Fuller, A fuzzy approach to real option valuation, Fuzzy Sets Syst. 139 (2003) 297–312. [2] C. Carlsson, R. Fuller, M. Heikkila, P. Majlender, A fuzzy approach to R&D project portfolio selection, Int. J. Approx. Reason. 44 (2007) 93–105. [3] P. Carr, The valuation of sequential exchange opportunities, J. Finance 43 (5) (1988) 1235–1256. [4] M. Collan, R. Fullér, J. Mezei, A fuzzy pay-off method for real option valuation, J. Appl. Math. Decis. Sci. 2009 (2009). [5] T. Copeland, V. Antikarov, Real Options: A Practitioner’s Guide, Texere, New York, 2003. [6] A.K. Dixit, Entry and exit decisions under uncertainty, J. Polit. Econ. 97 (1989) 620–638. [7] D. Dubois, H. Prade, The three semantics of fuzzy sets, Fuzzy Sets Syst. 90 (1997) 141–150. [8] D. Dubois, H. Prade, Possibility theory and its applications: where do we stand?, in: J. Kacprzyk, W. Pedrycz (Eds.), Springer Handbook of Computational Intelligence, Springer, 2015, pp. 31–60. [9] R. Geske, H.E. Johnson, The American put option valued analytically, J. Finance 39 (5) (1984) 1511–1524. [10] M.L. Guerra, L. Sorini, L. Stefani, Option price sensitivities through fuzzy numbers, Comput. Math. Appl. 61 (3) (2011) 515–526. [11] J.W. Lee, D.A. Paxson, Confined exponential approximations for the valuation of American options, Eur. J. Finance 9 (2003) 449–474. [12] J.W. Lee, D.A. Paxson, Valuation of R&D real American sequential exchange options, R & D Manag. 31 (2) (2001) 191–201. [13] W. Margrabe, The value of an exchange option to exchange one asset for another, J. Finance 33 (1) (1978) 177–186. [14] R.L. McDonald, D.R. Siegel, Investment and the valuation of firms when there is an option to shut down, Int. Econ. Rev. 28 (2) (1985) 331–349. [15] M.L. Puri, D. Ralescu, Fuzzy random variables, J. Math. Anal. Appl. 114 (1986) 409–422. [16] X. Wang, J. He, S. Li, Compound option pricing under fuzzy environment, J. Appl. Math. 2014 (2014) 1–9. [17] Y. Yoshida, M. Yasuda, J. Nakagami, M. Kurano, A new evaluation of mean value for fuzzy numbers and its application to American put option under uncertainty, Fuzzy Sets Syst. 157 (2006) 2614–2626. [18] L.A. Zadeh, Fuzzy sets as a basis for a theory of possibility, Fuzzy Sets Syst. 1 (1978) 3–28.