Laerte Sorini

Parametric representation of fuzzy numbers and application to fuzzy calculus

We present several models to obtain simple parametric representations of the fuzzy numbers or intervals, based on the use of piecewise monotonic functions of different forms. The representations have the advantage of allowing flexible and easy to control shapes of the fuzzy numbers (we use the standard α-cuts setting, but also the membership functions are obtained immediately) and can be used directly to obtain error-controlled-approximations of the fuzzy calculus in terms of a finite set of parameters. The general setting is the Hermite-type interpolation, where the values and the slopes of the monotonic interpolators are given by appropriate parameters and the overall errors of the fuzzy computations can be controlled within a prefixed tolerance by eventually augmenting the total number of pieces (and of the parameters) by which the results are obtained. The representations are designed to model the lower and the upper extremal values of each α-cut (compact) intervals of the fuzzy numbers and are able to produce almost any possible configuration (differentiable, continuous or with a finite number of discontinuity points) by using parametric monotonic functions of different types. We show applications in the standard fuzzy calculus and we stress the generality and the applicability of the proposed representation to a large class of problems, including the numerical solution of fuzzy differential equations, the fuzzy linear regression and the stochastic extensions of the fuzzy mathematics. The proposed model is called the Lower–Upper representation and we denote the associated fuzzy numbers or intervals as LU-fuzzy.

Option price sensitivities through fuzzy numbers

The main motivation in using fuzzy numbers in finance lies in the need for modelling the uncertainty and vagueness that are implicit in many situations. However, the fuzzy approach should not be considered as a substitute for the probabilistic approach but rather as a complementary way to describe the model peculiarities. Here, we consider, in particular, the Black and Scholes model for option pricing, and we show that the fuzzification of some key parameters enables a sensitivity analysis of the option price with respect to the risk-free interest rate, the final value of the underlying stock price, the volatility, and also better forecasts (see Thavaneswaran et al. (2009) [12] for details). The sensitivities with respect to the variables of the model are represented by different letters of the Greek alphabet and they play an important role in the definition of the shape of the fuzzy option price.

Parametrized Fuzzy Numbers for Option Pricing

The growing interest, during the last years, in the managing of risk in financial markets has involved primarily the pricing models for derivatives. However some of these models seemed to be soon unsatisfactory due to the incapability to capture the relevant stylized facts of real markets. Many attempts of fuzzy models have been recently proposed in the literature, but they either have the disadvantage of requiring a large amount of computations (e.g. constrained optimization problems) or they suffer a relative rigidity in representing and capturing the shapes of the fuzzy quantities (data and/or results). The parametrized fuzzy numbers (LU-fuzzy representation) that we have introduced recently (in L. Stefanini, L.Sorini, M.L.Guerra, Fuzzy Sets and Systems, 2006) reveal their great flexibility and simplicity to model fuzzy financial quantities and information in option pricing (and other financial applications) and to perform quickly the required fuzzy calculus. An additional important advantage is that the overestimation effect, inherent to the use of the interval fuzzy arithmetic, is completely eliminated.

Approximation of Fuzzy Numbers by F-Transform

The fuzzy transform setting (F-transform) is proposed as a tool for representation and approximation of type-1 and type-2 fuzzy numbers; the inverse F-transform on appropriate fuzzy partition of the membership interval [0,1] is used to characterize spaces of fuzzy numbers in such a way that arithmetic operations are defined and expressed in terms of the F-transform of the results. A type-2 fuzzy number is represented as a particular fuzzy-valued function and it is expressed in terms of a two-dimensional F-transform where the first dimension represents the universe domain and the second dimension represents the membership domain. Operators on two dimensional F-transform are then proposed to approximate arithmetic operations with type 2 fuzzy numbers.

Some Parametric Forms for LR Fuzzy Numbers and LR Fuzzy Arithmetic

In this paper we show that the models for parametric representation of fuzzy numbers in the level-cuts setting can be used to model LR fuzzy numbers and LR fuzzy arithmetic. This extends the family of LR fuzzy numbers to a sequence of finite-dimensional subspaces, approximating the space of fuzzy numbers with increasing goodness. The basic arithmetic with parametric LR fuzzy numbers is illustrated in an algorithmic framework.

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

The paper is organized as follows: section 2 contains a brief description of the fuzzy calculus with the LU-fuzzy model and in section 3 we describe the detailed algorithms which implement the LU-fuzzy extension principle. Section 4 contains the description of the LU-fuzzy calculator and its illustration to the application in the Black and Scholes fuzzy option pricing.

Type-2 fuzzy numbers and operations by F-transform

The fuzzy transform setting (F-transform) is proposed as a tool for representation and approximation of type-2 fuzzy numbers; a type-2 fuzzy number is represented as a particular fuzzy-valued function and it is expressed in terms of a two-dimensional F-transform where the first dimension represents the universe domain and the second dimension represents the membership domain. Operators on two dimensional F-transform are then proposed to approximate arithmetic operations with type-2 fuzzy numbers.

A new approach to linear programming with interval costs

Linear Programming problems are solved in the present paper when uncertainty in costs is modelled through interval numbers; the comparison index based on the generalized Hukuhara difference is adopted to indicate the possible relative positions of two intervals and to chose the proper solution.

Value Function Computation in Fuzzy Models by Differential Evolution

In this paper, we show that the possibilistic mean values produce computation results that may differ in a nontrivial may from those obtained with the fuzzy extension principle. The evidence is carried out by comparing some examples derived from several models in finance.

Time Series Modeling Based on Fuzzy Transform

It is well known that smoothing is applied to better see patterns and underlying trends in time series. In fact, to smooth a data set means to create an approximating function that attempts to capture important features in the data, while leaving out noises. In this paper we choose, as an approximation function, the inverse fuzzy transform (introduced by Perfilieva in Fuzzy Sets Syst 157:993–1023, 2006 [3]) that is based on fuzzy partitioning of a closed real interval into fuzzy subsets. The empirical distribution we introduce can be characterized by its expectiles in a similar way as it is characterized by quantiles.