On fuzzy real option valuation Supported by the Waeno project TEKES 4068299.

1 a -α a b b+β
Figure 1: Trapezoidal fuzzy number. Definition 1.2. A fuzzy set A ∈ F is called trapezoidal fuzzy number with core [a, b], left width α and right width β if its membership function has the following form a−t if a − α ≤ t ≤ a 1− α 1 if a ≤ t ≤ b A(t) = t−b 1− if a ≤ t ≤ b + β β 0 otherwise [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 . (1)
Keywords: Option pricing; Possibilistic mean value; Possibility distributions; Possibilistic variance
TUCS Research Group Institute for Advanced Management Systems Research
In other words, a1 (γ ) denotes the left-hand side and a2 (γ ) denotes the right-hand side of the γ -cut. We shall use the notation [A]γ = [a1 (γ ), a2 (γ )]. The support of A is the open interval (a1 (0), a2 (0)). Fuzzy numbers can also be considered as possibility distributions [3]. If A ∈ F is a fuzzy number and x ∈ R a real number then A(x) can be interpreted as the degree of possiblity of the statement ’x is A’. 1
On fuzzy real option valuation
Supported by the Waeno project TEKES 40682/99.
Christer Carlsson
Institute for Advanced Management Systems Research, e-mail:christer.carlsson@abo.fi
A fuzzy subset A of a classical set X is called normal if there exists an x ∈ X such that A(x) = 1. Otherwise A is subnormal. An α-level set of a fuzzy set A of X is a non-fuzzy set denoted by [A]α and is defined by [A]α = {t ∈ X |A(t) ≥ α} if α > 0, if α = 0, cl(suppA)
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where cl(suppA) denotes the closure of the support of A. A fuzzy set A of X is called convex if [A]α is a convex subset of X , ∀α ∈ [0, 1].
In many situations people are only able to characterize numeric information imprecisely. For example, people use terms such as, about $5,000, near zero, or essentially bigger than $5,000. These are examples of what are called fuzzy numbers. Using the theory of fuzzy subsets we can represent these fuzzy numbers as fuzzy subsets of the set of real numbers. More exactly, Definition 1.1. 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 numbers will be denoted by F .
1
Possibilistic mean value and variance of fuzzy numbers
Fuzzy sets were intorduced by Zadeh [6] in 1965 to represent/manipulate data and information possessing nonstatistical uncertainties. Let X be a nonempty set. A fuzzy set A in X is characterized by its 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 . The value zero is used to represent complete non-membership, the value one is used to represent complete membership, and values in between are used to represent intermediate degrees of membership. Frequently we will write simply A(x) instead of µA (x). The family of all fuzzy (sub)sets in X is denoted by F (X ).
Robert Full´ er
Department of OR, E¨ otv¨ os Lor´ and University and Institute for Advanced Management Systems Research, e-mail: robert.fuller@abo.fi
TUCS
Let A be a fuzzy number. Then [A]γ is a closed convex (compact) subset of R for all γ ∈ [0, 1]. Let us introduce the notations a1 (γ ) = min[A]γ , a2 (γ ) = max[A]γ
Turku Centre for Computer Science TUCS Technical Report No 367 October 2000 ISBN 952-12-0722-1 ISSN 1239-1891