Maria Letizia Guerra

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.

Approximate fuzzy arithmetic operations using monotonic interpolations

Abstract We suggest the use of piecewise monotonic interpolations to approximate and represent a fuzzy number (or interval) and to derive a procedure to control the absolute error associated to the arithmetic operations ( + ,-, · , : ) between fuzzy numbers, in order to reduce the distance between the true result of the operation and its approximation. The monotonic functions are then used to define a parametric representation of a large class of fuzzy numbers having general shapes of the membership function and a simple and accurate procedure is introduced for the fuzzy arithmetic. Several computational experiments are given to show the good performance of the proposed procedure.

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.

Interval and fuzzy Average Internal Rate of Return for investment appraisal

In investment appraisal, uncertainty can be managed through intervals or fuzzy numbers because the arithmetical properties and the extension principle are well established and can be successfully applied in a rigorous way. We apply interval and fuzzy numbers to the Average Internal Rate of Return (AIRR), recently introduced for overcoming the problems of the traditional Internal Rate of Return (IRR). In the setting of interval and fuzzy arithmetic, we establish relations between the interim capitals invested, the profits and the cash flows, which are the ingredients of the AIRR and shed lights on the different ways uncertainty propagates depending on which variable is known and which one is derived. The relations between fuzzy AIRR and fuzzy Net Present Value are also investigated.

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.

A comparison index for interval ordering

The comparison of intervals, following several order relations, is relevant in interval linear programming methods to solve many real-life problems. The determination of coefficients as crisp values is practically impossible in reality, where data sources are often uncertain, vague and incomplete. The uncertainty can be modelled with coefficients that vary in intervals, making possible as decision making process that is common in the areas of soft sciences like social science, economics, finance and management studies. To help comparison of intervals according to different relevant order relations, a general comparison index is proposed and some properties are illustrated.

A comparison index for interval ordering based on generalized Hukuhara difference

Interval methods is one option for managing uncertainty in optimization problems and in decision management. The precise numerical estimation of coefficients may be meaningless in real-world applications, because data sources are often uncertain, vague and incomplete. In this paper we introduce a comparison index for interval ordering based on the generalized Hukuhara difference; we show that the new index includes the commonly used order relations proposed in literature. The definition of a risk measure guarantees the possibility to quantify a worst-case loss when solving maximization or minimization problems with intervals.

Average Rate of Return with Uncertainty

In investment appraisal, uncertainty can be managed through intervals or fuzzy numbers. The arithmetical properties and the extension principle are well established and can be successfully applied in a rigorous way. The investments ranking is preferably performed when the decision maker dispone of an interest.

Towards a Coherent Volatility Pricing Model: An Empirical Comparison

Many empirical analysis suggest that market prices dynamics are not well captured by Black and Scholes model. A valid generalization is attained by allowing volatility to change randomly and different approaches have been proposed in literature since the pioneering model by Hull and White [9].

A comparative simulation study for estimating diffusion coefficient

An efficient methodology of estimation of parameters in the diffusion coefficient of the stochastic differential equation (SDE) is presented in this work. The methodology is based on the concept of quadratic variation of a stochastic process and on some classical numerical tools such as spline quadrature and bisection method.