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By I. M. Yaglom, V. G. Boltyanskii

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N − 1, and fα−n = fα+n , + d− αi ≥ 0 and dαi ≤ 0 for any i = 0, . . , n, − − − + + if fαi = fαi+1 (or fα+i = fα+i+1 ), then d− αi = dαi+1 (or dαi = dαi+1 ). An LU-fuzzy number will be denoted by A = (A− , A+ ) and the set of all LUfuzzy numbers defined over a partition 0 = a0 < · · · < αn = 1 will be denoted by FLU (α0 , . . , αn ). 1 For details, we refer to [10,19]. Option Pricing with Fuzzy Parameters via Monte Carlo Simulation 29 Let D ⊆ Rm and g : D → R be a real function which has all partial derivatives on the domain D, ie.

Option Pricing with Fuzzy Parameters via Monte Carlo Simulation 29 Let D ⊆ Rm and g : D → R be a real function which has all partial derivatives on the domain D, ie. gxk (a1 , . . , am ) ∈ R for any (a1 , . . , am ) ∈ D and k = 1, . . , m. A general procedura showing how to extend the function g to a function g˜ : D → FLU (α0 , . . , αn ), where D ⊆ FLU (α0 , . . , αn )m is a suitable domain, can be formulated within the two following steps:2 1. Put m = {1, . . ,π(m) fα0 . . ,π(m) dα0 . .

M) Bk ), (8) where min (and analogously max) is defined by min a c , b d = a b if and only if a ≤ c or (a = c and b ≤ d). Example 1. One can simply check that 1. (A− , A+ ) + (B− , B+ ) = (A− + B− , A+ + B+ ), 2. k(A− , A+ ) = (kA− , kA+ ) and −k(A− , A+ ) = (−kA+ , −kA− ) for k ≥ 0, 3. exp [(A− , A+ )] = (exp[A− ], exp[A+ ]) with exp[A− ]αk = exp[fα−k ] exp[fα−k ]d− αk and exp[A+ ]αk = exp[fα+k ] exp[fα+k ]d+ αk , where the usual addition of matrices and the usual scalar multiplication are applied.

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