The arithmetic operations on fuzzy numbers are usually approached either by the use of the extension principle (in the domain of the membership function) or by the interval arithmetics (in the domain of the &#945; &#8722; cuts). The exact analytical fuzzy mathematics dates back from the early eighties and are outlined by Dubois and Prade (see [1]); the same authors have introduced the well known L-R model and the corresponding formulas for the fuzzy operations (see [2]). Very recent literature on fuzzy numbers is rich of contributions on the fuzzy arithmetic operations and the use of simple formulas to approximate them; an extensive survey and bibliography on fuzzy intervals is in [3]. We suggest in [4] the use of monotonic splines to approximate the fuzzy numbers, using several interpolation forms (monotonic rational interpolators and mixed cubic-exponential interpolator) and we derive a procedure to control the locations of the nodes so that the error of the approximation is controlled by the possible insertion of additional nodes into the piecewise interpolation. We see that, with only a few nodes, our approximations of fuzzy calculus maintain accurate results. The parametric LU representation of the fuzzy numbers allows a set of possible shapes (types of membership functions) that seems to be much wider than the well-known L-R framework. The paper is organized as follows: section 2 contains a brief description of the fuzzy calculus; in section 3 we describe the LU-fuzzy model and in section 4 we describe the detailed algorithms which implement the LU-fuzzy extension principle. Section 5 contains the description of the LU-fuzzy calculator

Guerra M.L., Stefanini L., Sorini L. (2006). A Parameterization of Fuzzy Numbers for Fuzzy Calculus and Application to the Fuzzy Black-Scholes Option Pricing. s.l : IEEE.

### A Parameterization of Fuzzy Numbers for Fuzzy Calculus and Application to the Fuzzy Black-Scholes Option Pricing

#### Abstract

The arithmetic operations on fuzzy numbers are usually approached either by the use of the extension principle (in the domain of the membership function) or by the interval arithmetics (in the domain of the α − cuts). The exact analytical fuzzy mathematics dates back from the early eighties and are outlined by Dubois and Prade (see [1]); the same authors have introduced the well known L-R model and the corresponding formulas for the fuzzy operations (see [2]). Very recent literature on fuzzy numbers is rich of contributions on the fuzzy arithmetic operations and the use of simple formulas to approximate them; an extensive survey and bibliography on fuzzy intervals is in [3]. We suggest in [4] the use of monotonic splines to approximate the fuzzy numbers, using several interpolation forms (monotonic rational interpolators and mixed cubic-exponential interpolator) and we derive a procedure to control the locations of the nodes so that the error of the approximation is controlled by the possible insertion of additional nodes into the piecewise interpolation. We see that, with only a few nodes, our approximations of fuzzy calculus maintain accurate results. The parametric LU representation of the fuzzy numbers allows a set of possible shapes (types of membership functions) that seems to be much wider than the well-known L-R framework. The paper is organized as follows: section 2 contains a brief description of the fuzzy calculus; in section 3 we describe the LU-fuzzy model and in section 4 we describe the detailed algorithms which implement the LU-fuzzy extension principle. Section 5 contains the description of the LU-fuzzy calculator
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Guerra M.L., Stefanini L., Sorini L. (2006). A Parameterization of Fuzzy Numbers for Fuzzy Calculus and Application to the Fuzzy Black-Scholes Option Pricing. s.l : IEEE.
Guerra M.L.; Stefanini L.; Sorini L.
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Utilizza questo identificativo per citare o creare un link a questo documento: `https://hdl.handle.net/11585/45292`
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