Special Issue "Advanced Methods in Mathematical Finance"

A special issue of Mathematics (ISSN 2227-7390). This special issue belongs to the section "Financial Mathematics".

Deadline for manuscript submissions: closed (31 August 2020).

Special Issue Editors

Prof. Dr. Krzysztof Piasecki
E-Mail
Guest Editor
Institute of Economy and Finance, WSB University in Poznań, ul. Powstańców Wielkopolskich 5, 61-895 Poznań, Poland
Interests: mathematics for fuzzy systems; quantified behavioral finance; operations research; financial mathematics
Special Issues, Collections and Topics in MDPI journals
Dr. Anna Łyczkowska-Hanćkowiak
E-Mail Website
Guest Editor
Institute of Finance, WSB University of Poznań, ul. Powstańców Wielkopolskich 5, 61-895 Poznań, Poland
Interests: quantified behavioral finance; imprecise in assessing capital markets; operational research; discrete mathematics; graph theory; theory of matroids

Special Issue Information

Dear Colleagues,

Striving for wealth has accompanied people since the dawn of time. Nowadays, it seems that stock exchange investments can be the means to wealth. It is well-known that success in stock exchange investments is not guaranteed only by the investor’s knowledge of the way the stock exchange market functions and how the general economy operates. Along with this knowledge, the other factor that determines success in stock exchange investments is the player’s investment strategy. Models of financial mathematics, classified as applied mathematics, are used to describe the financial market and to create investment strategies. Experience shows that strategies that allow a player to win indefinitely do not exist. Experience indicates that one of the vital factors that guarantees financial success is the use of new, well-created investment methods that are not known to other actors. This causes a specific arms race in financial markets. This race creates demand for new advanced methods of financial mathematics.

These advanced methods can be understood in terms of the modernisation of the classical models of financial mathematics that grew out of the economic grounds of classic logic. The desired direction of development is the extension of models, and the creation of possibilities to include other factors that influence the financial market in these models. In the case of financial mathematics, it is not beneficial to generalise models in a way that does not include new categories of events in financial markets or new economic theories. This is due to the fact that financial mathematics belongs to applied mathematics.

An essential direction for the development of economic theories is the inclusion of descriptions of economic and financial phenomena in terms of multi-valued logic. The consequence of this evolution of financial and economic theories is the creation of advanced methods of financial mathematics and fuzzy financial models. These methods, among other things, reflect the various developmental directions of fuzzy set theories. The main interest of financial mathematics focuses on such fuzzy models, which were created by mathematical deduction. The analysis of the fuzzy financial models created by mathematical induction belongs to the field of operations research.

We can successfully contrast classical economics with behavioural economics. This fact has caused the creation of advanced methods of financial mathematics dedicated to behavioural finance.

The purpose of this Special Issue is to establish a collection of articles that presents the advanced formal methods related to following directions of the development of financial mathematics:

  • classical financial methods in terms of classical logic
  • fuzzy financial models
  • behavioural finance models.

We will also take into consideration justified proposals of other significant directions of the development of financial mathematics. We are motivated by the overriding aim of indicating the connections between mathematical deduction and real finance.

Prof. Dr. Krzysztof Piasecki
Dr. Anna Łyczkowska-Hanćkowiak
Guest Editors

Manuscript Submission Information

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Keywords

  • Classical financial mathematics
  • Fuzzy models of financial mathematics
  • Behavioural models of financial mathematics
  • Evaluation of assets
  • Discounting
  • Portfolio analysis
  • Financial risk

Published Papers (15 papers)

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Research

Article
Forecasting U.S. Aggregate Stock Market Excess Return: Do Functional Data Analysis Add Economic Value?
Mathematics 2020, 8(11), 2042; https://doi.org/10.3390/math8112042 - 16 Nov 2020
Cited by 2 | Viewed by 640
Abstract
This paper analyzes the forecast performance of historical S&P500 and Dow Jones Industrial Average (DJIA) excess returns while using nonparametric functional data analysis (NP-FDA). The empirical results show that the NP-FDA forecasting strategy outperforms not only the the prevailing-mean model, but also the [...] Read more.
This paper analyzes the forecast performance of historical S&P500 and Dow Jones Industrial Average (DJIA) excess returns while using nonparametric functional data analysis (NP-FDA). The empirical results show that the NP-FDA forecasting strategy outperforms not only the the prevailing-mean model, but also the traditional univariate predictive regressions with standard predictors used in the literature and, most cases, also combination approaches that use all predictors jointly. In addition, our results clearly have important implications for investors, from an asset allocation perspective, a mean-variance investor realizes substantial economic gains. Indeed, our results show that NP-FDA is the only one individual model that can overcome the historical average forecasts for excess returns in statistically and economically significant manners for both S&P500 and DJIA during the entire period, NBER recession, and expansions periods. Full article
(This article belongs to the Special Issue Advanced Methods in Mathematical Finance)
Article
On Tail Dependence and Multifractality
Mathematics 2020, 8(10), 1767; https://doi.org/10.3390/math8101767 - 13 Oct 2020
Viewed by 644
Abstract
We study whether, and if yes then how, a varying auto-correlation structure in different parts of distributions is reflected in the multifractal properties of a dynamic process. Utilizing the quantile autoregressive process with Gaussian copula using three popular estimators of the generalized Hurst [...] Read more.
We study whether, and if yes then how, a varying auto-correlation structure in different parts of distributions is reflected in the multifractal properties of a dynamic process. Utilizing the quantile autoregressive process with Gaussian copula using three popular estimators of the generalized Hurst exponent, our Monte Carlo simulation study shows that such dynamics translate into multifractal dynamics of the generated series. The tail-dependence of the auto-correlations forms strong enough non-linear dependencies to be reflected in the estimated multifractal spectra and separated from the case of the standard auto-regressive process. With a quick empirical example from financial markets, we argue that the interaction is more important for the asymmetric tail dependence. In addition, we discuss and explain the often reported paradox of higher multifractality of shuffled series compared to the original financial series. In short, the quantile-dependent auto-correlation structures qualify as sources of multifractality and they are worth further theoretical examination. Full article
(This article belongs to the Special Issue Advanced Methods in Mathematical Finance)
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Article
On a Free Boundary Problem for American Options Under the Generalized Black–Scholes Model
Mathematics 2020, 8(9), 1563; https://doi.org/10.3390/math8091563 - 11 Sep 2020
Cited by 1 | Viewed by 828
Abstract
We consider the problem of pricing American options using the generalized Black–Scholes model. The generalized Black–Scholes model is a modified form of the standard Black–Scholes model with the effect of interest and consumption rates. In general, because the American option problem does not [...] Read more.
We consider the problem of pricing American options using the generalized Black–Scholes model. The generalized Black–Scholes model is a modified form of the standard Black–Scholes model with the effect of interest and consumption rates. In general, because the American option problem does not have an exact closed-form solution, some type of approximation is required. A simple numerical method for pricing American put options under the generalized Black–Scholes model is presented. The proposed method corresponds to a free boundary (also called an optimal exercise boundary) problem for a partial differential equation. We use a transformed function that has Lipschitz character near the optimal exercise boundary to determine the optimal exercise boundary. Numerical results indicating the performance of the proposed method are examined. Several numerical results are also presented that illustrate a comparison between our proposed method and others. Full article
(This article belongs to the Special Issue Advanced Methods in Mathematical Finance)
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Article
Optimization Parameters of Trading System with Constant Modulus of Unit Return
Mathematics 2020, 8(8), 1384; https://doi.org/10.3390/math8081384 - 18 Aug 2020
Cited by 1 | Viewed by 944
Abstract
The unit return is determined as the return in the quotation currency (QCR) per the unit of base exchange medium (BEM). The main purpose is to examine the applicability of a trading system with a constant modulus of unit return (CMUR). The CMUR [...] Read more.
The unit return is determined as the return in the quotation currency (QCR) per the unit of base exchange medium (BEM). The main purpose is to examine the applicability of a trading system with a constant modulus of unit return (CMUR). The CMUR system supports speculative operations related to the exchange rate, given as the BEM quotation per the QCR. Premises for investment decisions are based on knowledge about the quotation dynamics described by its binary representation. This knowledge is described by a prediction table containing the conditional probability distributions of exchange rate increments. Any prediction table depends on observation range. Financial effectiveness of any CMUR system is assessed in the usual way by interest rate and risk index based on Shannon entropy. The main aim of our paper is to present algorithms which may be used for selecting effective CMUR systems. Required unit return modulus and observation range are control parameters applied for management of CMUR systems. Optimal values of these parameters are obtained by implementation of the proposed algorithm. All formal considerations are illustrated by an extensive case study linked to gold trading. Full article
(This article belongs to the Special Issue Advanced Methods in Mathematical Finance)
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Article
A Nonlinear Technical Indicator Selection Approach for Stock Markets. Application to the Chinese Stock Market
Mathematics 2020, 8(8), 1301; https://doi.org/10.3390/math8081301 - 06 Aug 2020
Cited by 1 | Viewed by 831
Abstract
In this paper we present a combinatorial nonlinear technical indicator approach for the identification of appropriate combinations of stock technical indicators as inputs in non-linear models. This approach is illustrated with the example of Chinese stock indexes and 35 different stock technical indicators [...] Read more.
In this paper we present a combinatorial nonlinear technical indicator approach for the identification of appropriate combinations of stock technical indicators as inputs in non-linear models. This approach is illustrated with the example of Chinese stock indexes and 35 different stock technical indicators using neural networks as the chosen non-linear method. Stock market technical indicators can generate contradictory signals regarding the future performance of the stock analyzed. Furthermore, some non-linear methods, such as neural networks, can have poor generalization power when dealing with problems of high dimensionality due to the issue of local minima. Therefore, non-linear approaches that can identify appropriate combinations of input variables are of clear importance. It will be shown that the proposed approach, when using neural networks as classifiers, generates error rates lower than those using directly neural networks without dimensionality reduction. It will also be shown that merely increasing the number of neurons does not increase the accuracy. The approach proposed in this article is illustrated with an application to the stock market using neural networks but it could be applied to other fields and it can also be used with other non-linear techniques such as for instance support vector machines. Full article
(This article belongs to the Special Issue Advanced Methods in Mathematical Finance)
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Article
Economic Policy Uncertainty and Stock Market Spillovers: Case of Selected CEE Markets
Mathematics 2020, 8(7), 1077; https://doi.org/10.3390/math8071077 - 02 Jul 2020
Cited by 4 | Viewed by 1132
Abstract
Rising political and economic uncertainty over the world affects all participants on different markets, including stock markets. Recent research has shown that these effects are significant and should not be ignored. This paper estimates the spillover effects of shocks in the economic policy [...] Read more.
Rising political and economic uncertainty over the world affects all participants on different markets, including stock markets. Recent research has shown that these effects are significant and should not be ignored. This paper estimates the spillover effects of shocks in the economic policy uncertainty (EPU) index and stock market returns and risks for selected Central and Eastern European markets (Bulgaria, Czech Republic, Estonia, Hungary, Lithuania, Poland, Croatia, Slovakia and Slovenia). Based on rolling estimations of the vector autoregression (VAR) model and the Spillover Indices, detailed insights are obtained on the sources of shock spillovers between the variables in the system. Recommendations are given based on the results both for policymakers and international investors. The contribution of the paper consists of the dynamic estimation approach, alongside allowing for the feedback relationship between the variables of interest, as well as examining the mentioned spillovers for the first time for majority of the observed countries. Full article
(This article belongs to the Special Issue Advanced Methods in Mathematical Finance)
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Article
A Complex MCDM Procedure for the Assessment of Economic Development of Units at Different Government Levels
Mathematics 2020, 8(7), 1067; https://doi.org/10.3390/math8071067 - 02 Jul 2020
Cited by 6 | Viewed by 624
Abstract
Studies on the economic development of government units are among the key challenges for authorities at different levels and an issue often investigated by economists. In spite of a considerable interest in the issue, there is no standard procedure for the assessment of [...] Read more.
Studies on the economic development of government units are among the key challenges for authorities at different levels and an issue often investigated by economists. In spite of a considerable interest in the issue, there is no standard procedure for the assessment of economic development level of units at different levels of government (national, regional, sub-regional). This assessment needs a complex system of methods and techniques applicable to the various types of data. So, adequate methods must be used at each level. This paper proposes a complex procedure for a synthetic indicator. The units are assessed at different government levels. Each level (national, regional, and sub-regional) may be described with a particular type of variables. Set of data may include variables with a normal or near-normal distribution, a strong asymmetry or extreme values. The objective of this paper is to present the potential behind the application of a complex Multi-Criteria Decision Making (MCDM) procedure based on the tail selection method used in the Extreme Value Theory (EVT), i.e., Mean Excess Function (MEF) together with one of the most popular MCDM methods, namely the Technique for Order of Preference by Similarity to Ideal Solution (TOPSIS), to assess the economic development level of units at different government levels. MEF is helpful to identify extreme values of variables and limit their impact on the ranking of local administrative units (LAUs). TOPSIS is suitable in ranking units described with multidimensional data set. The study explored the use of two types of TOPSIS (classical and positional) depending on the type of variables. These approaches were used in the assessment of economic development level of LAUs at national, regional and sub-regional levels in Poland in 2017. Full article
(This article belongs to the Special Issue Advanced Methods in Mathematical Finance)
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Article
Analyzing the Causality and Dependence between Gold Shocks and Asian Emerging Stock Markets: A Smooth Transition Copula Approach
Mathematics 2020, 8(1), 120; https://doi.org/10.3390/math8010120 - 13 Jan 2020
Cited by 6 | Viewed by 1206
Abstract
This study aims to investigate the causality and dependence structure of gold shocks and Asian emerging stock markets. The positive and negative shocks of gold prices are quantified, and Granger causality-based Vector autoregressive and Copula approaches are employed to measure the causality and [...] Read more.
This study aims to investigate the causality and dependence structure of gold shocks and Asian emerging stock markets. The positive and negative shocks of gold prices are quantified, and Granger causality-based Vector autoregressive and Copula approaches are employed to measure the causality and contagion effect, respectively, between the positive and negative gold shocks and Asian emerging stock markets’ volatilities. In addition, the nonlinear link between gold and stock markets is of concern and this motivates us to propose a Smooth Transition Dynamic Copula that allows for the structural change in time-varying dependence between gold shocks and Asian stock markets’ volatilities. Several Copula families are also considered, and the best-fit Copula model is used to explain the correlation or contagion effects. The findings of the study show that there is some significant causality between gold shocks and Asian stock markets’ volatilities in some parts of the sample period. We also observe a stronger correlation during the global financial crisis when compared to the pre- and post-crisis periods. In addition, the tail dependence is found between Indian stock and negative gold shock and between Korean stock and negative gold shock, which indicated the existence of the risk contagion effects between gold and these two stock markets. Full article
(This article belongs to the Special Issue Advanced Methods in Mathematical Finance)
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Article
Preference Based Portfolio for Private Investors: Discrete Choice Analysis Approach
Mathematics 2020, 8(1), 30; https://doi.org/10.3390/math8010030 - 24 Dec 2019
Cited by 4 | Viewed by 2095
Abstract
Behavioral finance literature shows that in addition to Markowitz’s rate of return and risk, private investors consider various other stock features. This paper discusses the problem of determining investors’ preferences for portfolio selection criteria, as well as the problem of optimal portfolio determination [...] Read more.
Behavioral finance literature shows that in addition to Markowitz’s rate of return and risk, private investors consider various other stock features. This paper discusses the problem of determining investors’ preferences for portfolio selection criteria, as well as the problem of optimal portfolio determination from the investors’ point of view. The study primarily focuses on private investors who are interested in one-time investments rather than stock trading. We use a discrete choice analysis and hierarchical Bayes method to measure individual investors’ preferences, and a logit model to determine individual shares of preferences. We treat the share of preferences as the share of certain stocks in an optimal portfolio. The proposed methodology is illustrated by the example of companies whose stocks are traded on the Belgrade Stock Exchange. We measure respondents’ preferences for companies, preferences for return rates, riskiness of stocks, and dividend rates. The results of comparing the performance of the resulting portfolio with the efficient frontier obtained using Markowitz’s portfolio theory indicate its high efficiency, thus validating the proposed approach. Full article
(This article belongs to the Special Issue Advanced Methods in Mathematical Finance)
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Article
A Nonparametric Approach to Bond Portfolio Immunization
Mathematics 2019, 7(11), 1121; https://doi.org/10.3390/math7111121 - 16 Nov 2019
Cited by 1 | Viewed by 806
Abstract
We consider the problem of short term immunization of a bond-like obligation with respect to changes in interest rates using a portfolio of bonds. In the case that the zero-coupon yield curve belongs to a fixed low-dimensional manifold, the problem is widely known [...] Read more.
We consider the problem of short term immunization of a bond-like obligation with respect to changes in interest rates using a portfolio of bonds. In the case that the zero-coupon yield curve belongs to a fixed low-dimensional manifold, the problem is widely known as parametric immunization. Parametric immunization seeks to make the sensitivities of the hedged portfolio price with respect to all model parameters equal to zero. However, within a popular approach of nonparametric (smoothing spline) term structure estimation, parametric hedging is not applicable right away. We present a nonparametric approach to hedging a bond-like obligation allowing for a general form of the term structure estimator with possible smoothing. We show that our approach yields the standard duration based immunization in the limit when the amount of smoothing goes to infinity. We also recover the industry best practice approach of hedging based on key rate durations as another particular case. The hedging portfolio is straightforward to calculate using only basic linear algebra operations. Full article
(This article belongs to the Special Issue Advanced Methods in Mathematical Finance)
Article
Predicting Contagion from the US Financial Crisis to International Stock Markets Using Dynamic Copula with Google Trends
Mathematics 2019, 7(11), 1032; https://doi.org/10.3390/math7111032 - 02 Nov 2019
Cited by 3 | Viewed by 1182
Abstract
The accuracy of contagion prediction has been one of the most widely investigated and challenging problems in economic research. Much effort has been devoted to investigating the key determinant of contagion and enhancing more powerful prediction models. In this study, we aim to [...] Read more.
The accuracy of contagion prediction has been one of the most widely investigated and challenging problems in economic research. Much effort has been devoted to investigating the key determinant of contagion and enhancing more powerful prediction models. In this study, we aim to improve the prediction of the contagion effect from the US stock market to the international stock markets by utilizing Google Trends as a new leading indicator for predicting contagion. To improve this contagion prediction, the dynamic copula models are used to investigate the structure of dependence between international markets and the US market, before, during, and after the occurrence of the US financial crisis in 2008. We also incorporate the Google Trends data as the exogenous variables in the time-varying copula equation. Thus, the ARMAX process is introduced. To investigate the predictive power of Google Trends, we employ the likelihood ratio test. Our empirical findings support that Google Trends is a significant leading indicator for predicting contagion in seven out of 10 cases: SP-FTSE, SP-TSX, SP-DAX, SP-Nikkei, SP-BVSP, SP-SSEC, and SP-BSESN pairs. Our Google-based models seem to predict particularly well the effect of the US crisis in 2008. In addition, we find that the contribution of Google Trends to contagion prediction varies among the different stock market pairs. This finding leads to our observation that the more volatile the market time-varying correlation, the more useful Google Trends. Full article
(This article belongs to the Special Issue Advanced Methods in Mathematical Finance)
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Article
The Forex Trading System for Speculation with Constant Magnitude of Unit Return
Mathematics 2019, 7(7), 623; https://doi.org/10.3390/math7070623 - 12 Jul 2019
Cited by 6 | Viewed by 1380
Abstract
The main purpose of this article is to investigate a speculative trading system with a constant magnitude of return rate. We consider speculative operations related to the exchange rate given as the quotient of the base exchange medium by the quoted currency. An [...] Read more.
The main purpose of this article is to investigate a speculative trading system with a constant magnitude of return rate. We consider speculative operations related to the exchange rate given as the quotient of the base exchange medium by the quoted currency. An exchange medium is understood as any currency or any precious metal. The unit return is defined as the return expressed in the quoted currency by the amount of base exchange medium. All possible states of the exchange market form a finite elemental space. All knowledge about the dynamics of this market is presented as a prediction table describing the conditional probability distributions of incoming exchange rate changes. On the other hand, in the proposed trading system each speculative operation is concluded in such a way that the gross payment is determined by the given magnitude of unit return. The paper contains an analysis of the following evaluation criteria: annual number of transaction, success probability, expected unit payment, expected unit profit, risk index, unit risk premium, return rate, interest rate, and interest risk premium. Both of these indices can be used to select the effective trading systems. Effectiveness is considered in the local sense and in the global sense. Full article
(This article belongs to the Special Issue Advanced Methods in Mathematical Finance)
Article
Risk Measurement of Stock Markets in BRICS, G7, and G20: Vine Copulas versus Factor Copulas
Mathematics 2019, 7(3), 274; https://doi.org/10.3390/math7030274 - 18 Mar 2019
Cited by 8 | Viewed by 1253
Abstract
Multivariate copulas have been widely used to handle risk in the financial market. This paper aimed to adopt two novel multivariate copulas, Vine copulas and Factor copulas, to measure and compare the financial risks of the emerging economy, developed economy, and global economy. [...] Read more.
Multivariate copulas have been widely used to handle risk in the financial market. This paper aimed to adopt two novel multivariate copulas, Vine copulas and Factor copulas, to measure and compare the financial risks of the emerging economy, developed economy, and global economy. In this paper, we used data from three groups (BRICS, which stands for emerging markets, specifically, those of Brazil, Russia, India, China, and South Africa; G7, which refers to developed countries; and G20, which represents the global market), separated into three periods (pre-crisis, crisis, and post-crisis) and weighed Value at Risk (VaR) and Expected Shortfall (ES) (based on their market capitalization) to compare among three copulas, C-Vine, D-Vine, and Factor copulas. Also, real financial data demonstrated that Factor copulas have stronger stability and perform better than the other two copulas in high-dimensional data. Moreover, we showed that BRICS has the highest risk and G20 has the lowest risk of the three groups. Full article
(This article belongs to the Special Issue Advanced Methods in Mathematical Finance)
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Article
Sharpe’s Ratio for Oriented Fuzzy Discount Factor
Mathematics 2019, 7(3), 272; https://doi.org/10.3390/math7030272 - 16 Mar 2019
Cited by 7 | Viewed by 890
Abstract
The analysis presented in this paper regards the security of a present value given as an ordered fuzzy number. The present value was estimated in an imprecise manner and supplemented by the forecast of its coming changes. A discount factor of such security [...] Read more.
The analysis presented in this paper regards the security of a present value given as an ordered fuzzy number. The present value was estimated in an imprecise manner and supplemented by the forecast of its coming changes. A discount factor of such security is an ordered fuzzy number of the orientation identical to the oriented present value that determines it. All classical methods of portfolio analysis are based on the definition of the return rate. In the case of securities with a fuzzy present value, a discount factor is a better tool for portfolio analysis than the return rate, which implies the chosen methods of management of securities should be revised and transformed to equivalent methods based on a discount factor. This would enable the use of those methods in the case of a financial instrument of the oriented fuzzy present value. This paper presents example results of the realization of such a postulate. The main aim of the paper is to generalize Sharpe’s ratio to a case of investment recommendations management formulated for a security characterized by an oriented discount factor. A five-degree rating scale was used. The whole deliberation is illustrated by broad numerical examples. Full article
(This article belongs to the Special Issue Advanced Methods in Mathematical Finance)
Article
Evaluation of the Impact of Strategic Offers on the Financial and Strategic Health of the Company—A Soft System Dynamics Approach
Mathematics 2019, 7(2), 208; https://doi.org/10.3390/math7020208 - 24 Feb 2019
Cited by 6 | Viewed by 1250
Abstract
When analyzing the possibility of supporting the decision-making process, one should take into account the essential properties of economic entities (the system and its objects). As a result, the development of an effective business model ought to be based on rationality and the [...] Read more.
When analyzing the possibility of supporting the decision-making process, one should take into account the essential properties of economic entities (the system and its objects). As a result, the development of an effective business model ought to be based on rationality and the characteristics of the system being modeled. Such an approach implies the use of an appropriate analysis and modeling method. Since the majority of relationships in the model are described using the experts’ tacit knowledge, methods known as “soft” are more suitable than “hard” in those situations. Fuzzy cognitive mappings (FCM) are therefore commonly used as a technique for participatory modeling of the system, where stakeholders can convey their knowledge to the model of the system in question. In this study, we introduce a novel approach: the extended weighted influence nonlinear gauge system (WINGS), which may equally well be applied to the decision problems of this type. Appraisal of high-value and long-term offers in the sector of the telecommunication supplier industry serves as a real-world case study for testing the new method. A comparison with FCM provides a deeper understanding of the similarities and differences of the two approaches. Full article
(This article belongs to the Special Issue Advanced Methods in Mathematical Finance)
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