Laerte Sorini

Urbino "Carlo Bo", Dipartimento di Economia, Società, Politica – DESP, Faculty Member

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Papers by Laerte Sorini

On the approximation of a membership function by empirical quantile functions

International Journal of Approximate Reasoning, Sep 1, 2020

Abstract The Average Cumulative representation of fuzzy intervals is connected with the possibili... more Abstract The Average Cumulative representation of fuzzy intervals is connected with the possibility theory in the sense that the possibility and necessity functions are substituted by a pair of non decreasing functions defined as the positive and negative variations in the Jordan decomposition of a membership function. In this paper we motivate the crucial role of ACF in determining the membership function from experimental data; some examples and simulations are shown to state the robustness of the proposed construction.

Testing robustness in calibration of stochastic volatility models

European Journal of Operational Research, May 1, 2005

Many numerical aspects are involved in parameter estimation of stochastic volatility models. We i... more

An Algorithm for Step Control in Numerical Solution of SDE

IFAC Proceedings Volumes, Jun 1, 1998

The explicit solution of a Stochastic Differential Equation (SDE) can be obtained only when the d... more The explicit solution of a Stochastic Differential Equation (SDE) can be obtained only when the drift and diffusion coefficients are linear. However a qualitative analysis can be done introducing numerical schemes for the approximation of SDEs solutions. In particular an algorithm for step control in numerical solution of SDE is presented here. Numerical solution of SDEs is applied to a large number of research's fields, among these finance is now one of the most interesting. In fact the dynamics of stock prices, interest rates, volatilities and other financial instruments are often supposed to be driven by SDEs. In the classical schemes of Euler and Milstein and in the more recent one due to Kloeden and Platen the discretization step is assumed to be constant. The approximation error becomes smaller as the length of step decreases but a too much small step implies higher computational time and this is not a desirable feature. The algorithm presented reduces the discretization step only when the trajectory shows quick variations and so the approximation error is reduced without increasing too much the number of iterations. Copyright ~ 1998lFAC

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Interval Analysis and Calculus for Interval-Valued Functions of a Single Variable—Part II: Extremal Points, Convexity, Periodicity

Axioms, Oct 14, 2019

We continue the presentation of new results in the calculus for interval-valued functions of a si... more We continue the presentation of new results in the calculus for interval-valued functions of a single real variable. We start here with the results presented in part I of this paper, namely, a general setting of partial orders in the space of compact intervals (in midpoint-radius representation) and basic results on convergence and limits, continuity, gH-differentiability, and monotonicity. We define different types of (local) minimal and maximal points and develop the basic theory for their characterization. We then consider some interesting connections with applied geometry of curves and the convexity of interval-valued functions is introduced and analyzed in detail. Further, the periodicity of interval-valued functions is described and analyzed. Several examples and pictures accompany the presentation.

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Quantile and expectile smoothing based on L1-norm and L2-norm fuzzy transforms

International Journal of Approximate Reasoning, Apr 1, 2019

The fuzzy transform (F-transform), introduced by I. Perfilieva, is a powerful tool for the constr... more The fuzzy transform (F-transform), introduced by I. Perfilieva, is a powerful tool for the construction of fuzzy approximation models; it is based on generalized fuzzy partitions and it is obtained by minimizing a quadratic (L 2-norm) error function. In this paper, within the discrete setting, we describe an analogous construction by minimizing an L 1-norm error function, so obtaining the L 1-norm F-transform, which is again a general approximation tool. The L 1-norm and L 2-norm settings are then used to construct two types of fuzzyvalued F-transforms, by defining expectile (L 2-norm) and quantile (L 1-norm) extensions of the transforms. This allows to model an observed time series in terms of fuzzy-valued functions, whose level-cuts can be interpreted in the setting of expectile and quantile regression. The proposed methodology is illustrated on some financial daily time series.

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Value function computation in fuzzy models by differential evolution

RePEc: Research Papers in Economics, 2013

In this paper, we show that the possibilistic mean values produce computation results that may di... more

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Quantile and expectile smoothing by F-trasform

Incorporating uncertainty in financial models

ABSTRACT Financial problems are often based on a rigorous mathematical struc-ture and are strongl... more ABSTRACT Financial problems are often based on a rigorous mathematical struc-ture and are strongly dependent on the imprecision of the input data. In this paper we show how it is possible to model the uncertainty by assuming that the fundamental parameters are fuzzy numbers. The massive computation effort involved in computations, can be success-fully handled due to a flexible and precise parametric representation of the lower and upper branches of the membership functions of the fuzzy numbers.

Variance reduction in a stochastic volatility scenario

This paper investigates the use of a variance reduction, called importance sampling, for Monte Ca... more This paper investigates the use of a variance reduction, called importance sampling, for Monte Carlo methods in the case of the stochastic volatility model for option pricing introduced by Hobson and Rogers (1998). We briefly recall that a European call option contract gives the right, but not the obligation, to buy a specific amount of a given stock or index at a specified price (strike price) in a specified time (maturity); we show some evidence on the call options on MIB30 Italian Index to verify the performance of the importance sampling in a complete stochastic volatility model. In Monte Carlo method the price of a call option is obtained as the average value of the simulations of a large number of independent, uniform variates (prices) by means of pseudo-random number generators. It is shown, finally, that variance is dramatically reduced meaning that numerical techniques introduced for variance reduction have still a lot to say.

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A Desktop Calculator for Parametric Fuzzy Arithmetic

European Society for Fuzzy Logic and Technology Conference, 2009

About three decades ago, Dubois and Prade developed the arithmetical structure of fuzzy numbers a... more

Reducing Variance in Hobson and Rogers Model for Option Pricing

Value at Risk Based on Fuzzy Numbers

Axioms, Aug 12, 2020

Value at Risk (VaR) has become a crucial measure for decision making in risk management over the ... more Value at Risk (VaR) has become a crucial measure for decision making in risk management over the last thirty years and many estimation methodologies address the finding of the best performing measure at taking into account unremovable uncertainty of real financial markets. One possible and promising way to include uncertainty is to refer to the mathematics of fuzzy numbers and to its rigorous methodologies which offer flexible ways to read and to interpret properties of real data which may arise in many areas. The paper aims to show the effectiveness of two distinguished models to account for uncertainty in VaR computation; initially, following a non parametric approach, we apply the Fuzzy-transform approximation function to smooth data by capturing fundamental patterns before computing VaR. As a second model, we apply the Average Cumulative Function (ACF) to deduce the quantile function at point p as the potential loss VaR p for a fixed time horizon for the 100p% of the values. In both cases a comparison is conducted with respect to the identification of VaR through historical simulation: twelve years of daily S&P500 index returns are considered and a back testing procedure is applied to verify the number of bad VaR forecasting in each methodology. Despite the preliminary nature of the research, we point out that VaR estimation, when modelling uncertainty through fuzzy numbers, outperforms the traditional VaR in the sense that it is the closest to the right amount of capital to allocate in order to cover future losses in normal market conditions.

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Uncertain parameters as fuzzy numbers in option pricing models

In this paper we show that the so called fuzzy--stochastic approach in financial models is an eff... more In this paper we show that the so called fuzzy--stochastic approach in financial models is an efficient way to handle the uncertainty about parameters. We show the possible applications in the option pricing models with a constant and stochastic volatility. Fuzzy numbers, extension principle, sensitivity analysis, fuzzy stochastic approach in financial models, option pricing models

A comparison index for linear programming models with interval costs

RePEc: Research Papers in Economics, 2018

Interval linear programming (ILP) has a long history but it still reserves many interesting sugge... more Interval linear programming (ILP) has a long history but it still reserves many interesting suggestions when it is applied to model real problems that are indivisible by uncertainty. In this paper we apply the comparison index for interval ordering based on the generalized Hukuhara difference to compare solutions in the ILP and we analyse the robustness of our methodology.

Quantile and expectile smoothing by F-trasform

QUADERNI DI ECONOMIA, MATEMATICA E STATISTICA, 2015

Reducing Variance in Hobson and Rogers Model for Option Pricing

A new approach to linear programming with interval costs

2017 IEEE International Conference on Fuzzy Systems (FUZZ-IEEE), 2017

Linear Programming problems are solved in the present paper when uncertainty in costs is modelled... more

Interval Analysis and Calculus for Interval-Valued Functions of a Single Variable. Part I: Partial Orders, gH-Derivative, Monotonicity

Axioms, 2019

We present new results in interval analysis (IA) and in the calculus for interval-valued function... more We present new results in interval analysis (IA) and in the calculus for interval-valued functions of a single real variable. Starting with a recently proposed comparison index, we develop a new general setting for partial order in the (semi linear) space of compact real intervals and we apply corresponding concepts for the analysis and calculus of interval-valued functions. We adopt extensively the midpoint-radius representation of intervals in the real half-plane and show its usefulness in calculus. Concepts related to convergence and limits, continuity, gH-differentiability and monotonicity of interval-valued functions are introduced and analyzed in detail. Graphical examples and pictures accompany the presentation. A companion Part II of the paper will present additional properties (max and min points, convexity and periodicity).

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Step control in numerical solution of stochastic differential equations

L'impiego di giochi interattivi nei processi di apprendimento

Obiettivo di questo lavoro è di presentare la creazione, l'esecuzioe e i risultati di un gioco in... more Obiettivo di questo lavoro è di presentare la creazione, l'esecuzioe e i risultati di un gioco interattivo (Multicountry) cercando di metterne in evidenza le potenzialità di apprendimento matematico-quantitativo dell'economia, e le possibilità di produrre artificialmente informazioni quantitative che possono essere proficuamente utilizzate negli studi di economia sperimentale. Il gioco è basato sul software (Macrosim) sviluppato presso la UW che può essere applicato in diversi contesti (macroeconomia, industria bancaria, contratti futures). Dai risultati del gioco, come ci saremmo aspettati, emerge come gli studenti attraverso esso riescano gradualmente ad apprendere il funzionamento della loro economia, di come quest'ultima reagisca alle loro decisioni e a quelle degli altri giocatori

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