{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,7,22]],"date-time":"2026-07-22T21:23:03Z","timestamp":1784755383761,"version":"3.55.0"},"reference-count":79,"publisher":"MDPI AG","issue":"8","license":[{"start":{"date-parts":[[2024,7,26]],"date-time":"2024-07-26T00:00:00Z","timestamp":1721952000000},"content-version":"vor","delay-in-days":0,"URL":"https:\/\/creativecommons.org\/licenses\/by\/4.0\/"}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Axioms"],"abstract":"<jats:p>Since the early 21st century, within fuzzy mathematics, there has been a stream of research in the field of option pricing that introduces vagueness in the parameters governing the movement of the underlying asset price through fuzzy numbers (FNs). This approach is commonly known as fuzzy random option pricing (FROP). In discrete time, most contributions use the binomial groundwork with up-and-down moves proposed by Cox, Ross, and Rubinstein (CRR), which introduces epistemic uncertainty associated with volatility through FNs. Thus, the present work falls within this stream of literature and contributes to the literature in three ways. First, analytical developments allow for the introduction of uncertainty with intuitionistic fuzzy numbers (IFNs), which are a generalization of FNs. Therefore, we can introduce bipolar uncertainty in parameter modelling. Second, a methodology is proposed that allows for adjusting the volatility with which the option is valued through an IFN. This approach is based on the existing developments in the literature on adjusting statistical parameters with possibility distributions via historical data. Third, we introduce into the debate on fuzzy random binomial option pricing the analytical framework that should be used in modelling upwards and downwards moves. In this sense, binomial modelling is usually employed to value path-dependent options that cannot be directly evaluated with the Black\u2013Scholes\u2013Merton (BSM) model. Thus, one way to assess the suitability of binomial moves for valuing a particular option is to approximate the results of the BSM in a European option with the same characteristics as the option of interest. In this study, we compared the moves proposed by Renddleman and Bartter (RB) with CRR. We have observed that, depending on the moneyness degree of the option and, without a doubt, on options traded at the money, RB modelling offers greater convergence to BSM prices than does CRR modelling.<\/jats:p>","DOI":"10.3390\/axioms13080503","type":"journal-article","created":{"date-parts":[[2024,7,26]],"date-time":"2024-07-26T13:04:59Z","timestamp":1721999099000},"page":"503","update-policy":"https:\/\/doi.org\/10.3390\/mdpi_crossmark_policy","source":"Crossref","is-referenced-by-count":4,"title":["Modelling Up-and-Down Moves of Binomial Option Pricing with Intuitionistic Fuzzy Numbers"],"prefix":"10.3390","volume":"13","author":[{"ORCID":"https:\/\/orcid.org\/0000-0002-7715-779X","authenticated-orcid":false,"given":"Jorge de","family":"Andr\u00e9s-S\u00e1nchez","sequence":"first","affiliation":[{"name":"Social and Business Research Lab., Universitat Rovira i Virgili, Campus de Bellissens, 43204 Reus, Spain"}],"role":[{"vocabulary":"crossref","role":"author"}]}],"member":"1968","published-online":{"date-parts":[[2024,7,26]]},"reference":[{"key":"ref_1","doi-asserted-by":"crossref","first-page":"637","DOI":"10.1086\/260062","article-title":"The pricing of options and corporate liabilities","volume":"81","author":"Black","year":"1973","journal-title":"J. Political Econ."},{"key":"ref_2","doi-asserted-by":"crossref","first-page":"141","DOI":"10.2307\/3003143","article-title":"Theory of rational option pricing","volume":"4","author":"Merton","year":"1973","journal-title":"Bell J. Econ. Manag. Sci."},{"key":"ref_3","unstructured":"Hull, J.C. (2008). Options Futures and Other Derivatives, Pearson Education."},{"key":"ref_4","doi-asserted-by":"crossref","unstructured":"Van der Hoek, J., and Elliott, R.J. (2006). Binomial Models in Finance, Springer.","DOI":"10.1007\/0-387-31607-8"},{"key":"ref_5","first-page":"38","article-title":"A Synthesis of Binomial Option Pricing Models for Lognormally Distributed Assets","volume":"18","author":"Chance","year":"2008","journal-title":"J. Appl. Financ. (Former. Financ. Pract. Educ.)"},{"key":"ref_6","doi-asserted-by":"crossref","first-page":"229","DOI":"10.1016\/0304-405X(79)90015-1","article-title":"Option Pricing: A Simplified Approach","volume":"7","author":"Cox","year":"1979","journal-title":"J. Financ. Econ."},{"key":"ref_7","first-page":"1092","article-title":"Two state option pricing","volume":"34","author":"Rendleman","year":"1979","journal-title":"J. Financ."},{"key":"ref_8","doi-asserted-by":"crossref","first-page":"309","DOI":"10.2307\/2331209","article-title":"A log-transformed binomial numerical analysis method for valuing complex multioption investments","volume":"26","author":"Trigeorgis","year":"1991","journal-title":"J. Financ. Quant. Anal."},{"key":"ref_9","first-page":"987","article-title":"Two-state option pricing: Binomial models revisited","volume":"21","author":"Jabbour","year":"2001","journal-title":"J. Futures Mark. Futures Options Other Deriv. Prod."},{"key":"ref_10","doi-asserted-by":"crossref","first-page":"552","DOI":"10.1016\/j.cam.2007.12.006","article-title":"On theoretical pricing of options with fuzzy estimators","volume":"223","author":"Chrysafis","year":"2009","journal-title":"J. Comput. Appl. Math."},{"key":"ref_11","doi-asserted-by":"crossref","first-page":"297","DOI":"10.1016\/S0165-0114(02)00591-2","article-title":"A fuzzy approach to real option valuation","volume":"139","author":"Carlsson","year":"2003","journal-title":"Fuzzy Sets Syst."},{"key":"ref_12","doi-asserted-by":"crossref","first-page":"119868","DOI":"10.1016\/j.eswa.2023.119868","article-title":"A systematic review of the interactions of fuzzy set theory and option pricing","volume":"223","year":"2023","journal-title":"Expert Syst. Appl."},{"key":"ref_13","doi-asserted-by":"crossref","first-page":"392","DOI":"10.1109\/TFUZZ.2016.2574906","article-title":"Fuzzy approaches to option price modelling","volume":"25","author":"Muzzioli","year":"2016","journal-title":"IEEE Trans. Fuzzy Syst."},{"key":"ref_14","doi-asserted-by":"crossref","first-page":"5883","DOI":"10.1007\/s00500-018-3251-5","article-title":"Interval-based, nonparametric approach for resampling of fuzzy numbers","volume":"23","author":"Romaniuk","year":"2019","journal-title":"Soft Comput."},{"key":"ref_15","doi-asserted-by":"crossref","first-page":"303","DOI":"10.1016\/S0377-2217(01)00042-X","article-title":"Application of the fuzzy-stochastic methodology to appraising the firm value as a European call option","volume":"135","author":"Zmeskal","year":"2001","journal-title":"Eur. J. Oper. Res."},{"key":"ref_16","doi-asserted-by":"crossref","first-page":"1069","DOI":"10.1016\/S0305-0548(03)00065-0","article-title":"Pricing European options based on the fuzzy pattern of Black-Scholes formula","volume":"31","author":"Wu","year":"2004","journal-title":"Comput. Oper. Res."},{"key":"ref_17","doi-asserted-by":"crossref","first-page":"515","DOI":"10.1016\/j.camwa.2010.11.024","article-title":"Option price sensitivities through fuzzy numbers","volume":"61","author":"Guerra","year":"2011","journal-title":"Comput. Math. Appl."},{"key":"ref_18","doi-asserted-by":"crossref","first-page":"1888","DOI":"10.1007\/s40815-018-0493-4","article-title":"Optimal Investment Timing with Investment Propensity Using Fuzzy Real Options Valuation","volume":"20","author":"Kim","year":"2018","journal-title":"Int. J. Fuzzy Syst."},{"key":"ref_19","doi-asserted-by":"crossref","first-page":"1659","DOI":"10.1142\/S0219622018500335","article-title":"Indices for Financial Market Volatility Obtained Through Fuzzy Regression","volume":"17","author":"Muzzioli","year":"2018","journal-title":"Int. J. Inf. Technol. Decis. Mak."},{"key":"ref_20","doi-asserted-by":"crossref","first-page":"211","DOI":"10.1007\/s10700-020-09316-x","article-title":"Option implied moments obtained through fuzzy regression","volume":"19","author":"Muzzioli","year":"2020","journal-title":"Fuzzy Optim. Decis. Mak."},{"key":"ref_21","doi-asserted-by":"crossref","first-page":"1","DOI":"10.1016\/j.fss.2019.04.027","article-title":"SMART-or and SMART-and fuzzy average operators: A generalized proposal","volume":"395","author":"Capotorti","year":"2020","journal-title":"Fuzzy Sets Syst."},{"key":"ref_22","doi-asserted-by":"crossref","first-page":"143","DOI":"10.1016\/j.fss.2022.09.007","article-title":"Cooperative R&D investment decisions: A fuzzy real option approach","volume":"458","author":"Anzilli","year":"2023","journal-title":"Fuzzy Sets Syst."},{"key":"ref_23","doi-asserted-by":"crossref","first-page":"1843","DOI":"10.1587\/transinf.2017EDP7236","article-title":"Fuzzy Levy-GJR-GARCH American option pricing model based on an infinite pure jump process","volume":"E101D","author":"Zhang","year":"2018","journal-title":"IEICE Trans. Inf. Syst."},{"key":"ref_24","doi-asserted-by":"crossref","unstructured":"Nowak, P., and Paw\u0142owski, M. (2023). Application of the Esscher Transform to Pricing Forward Contracts on Energy Markets in a Fuzzy Environment. Entropy, 25.","DOI":"10.3390\/e25030527"},{"key":"ref_25","doi-asserted-by":"crossref","first-page":"153","DOI":"10.1016\/S0377-2217(02)00591-X","article-title":"A discrete-time model of American put option in an uncertain environment","volume":"151","author":"Yoshida","year":"2003","journal-title":"Eur. J. Oper. Res."},{"key":"ref_26","doi-asserted-by":"crossref","first-page":"861","DOI":"10.1016\/S0165-1889(03)00060-5","article-title":"A multiperiod binomial model for pricing options in a vague world","volume":"28","author":"Muzzioli","year":"2004","journal-title":"J. Econ. Dyn. Control"},{"key":"ref_27","doi-asserted-by":"crossref","first-page":"255","DOI":"10.1007\/s11156-005-4767-1","article-title":"A fuzzy set approach for generalized CRR model: An empirical analysis of S&P 500 index options","volume":"25","author":"Lee","year":"2005","journal-title":"Rev. Quant. Financ. Account."},{"key":"ref_28","doi-asserted-by":"crossref","first-page":"3548","DOI":"10.1109\/TFUZZ.2021.3118781","article-title":"Fuzzy Option Pricing Based on Fuzzy Number Binary Tree Model","volume":"30","author":"Wang","year":"2022","journal-title":"IEEE Trans. Fuzzy Syst."},{"key":"ref_29","first-page":"47","article-title":"A study of european fuzzy put option buyers model on future contracts involving general trapezoidal fuzzy numbers","volume":"8","author":"Meenakshi","year":"2021","journal-title":"Glob. Stoch. Anal."},{"key":"ref_30","first-page":"331","article-title":"On some properties of American fuzzy put option model on fuzzy future contracts involving general linear octagonal fuzzy numbers","volume":"21","author":"Meenakshi","year":"2021","journal-title":"Adv. Appl. Math. Sci."},{"key":"ref_31","doi-asserted-by":"crossref","first-page":"1096","DOI":"10.1016\/j.ejor.2010.05.045","article-title":"Generalized soft binomial American real option pricing model (fuzzy-stochastic approach)","volume":"207","author":"Zmeskal","year":"2010","journal-title":"Eur. J. Oper. Res."},{"key":"ref_32","doi-asserted-by":"crossref","first-page":"15296","DOI":"10.1016\/j.eswa.2011.06.010","article-title":"A fuzzy real option approach for investment","volume":"38","author":"Ho","year":"2011","journal-title":"Expert Syst. Appl."},{"key":"ref_33","doi-asserted-by":"crossref","first-page":"578","DOI":"10.1016\/j.ins.2017.10.001","article-title":"Pricing of minimum guarantees in life insurance contracts with fuzzy volatility","volume":"460","author":"Anzilli","year":"2018","journal-title":"Inf. Sci."},{"key":"ref_34","doi-asserted-by":"crossref","first-page":"122","DOI":"10.1057\/s41278-021-00201-x","article-title":"Assessment of investment decisions in bulk shipping through fuzzy real options analysis","volume":"25","author":"Zhang","year":"2023","journal-title":"Marit. Econ. Logist."},{"key":"ref_35","first-page":"34","article-title":"Binomial tree model of the European option pricing based on the triangular intuitionistic fuzzy numbers","volume":"33","author":"Zhang","year":"2013","journal-title":"Syst. Eng. Theory Pract."},{"key":"ref_36","doi-asserted-by":"crossref","first-page":"116388","DOI":"10.1016\/j.eswa.2021.116388","article-title":"Generalized soft multimode real options model (fuzzy-stochastic approach)","volume":"192","author":"Zmeskal","year":"2022","journal-title":"Expert Syst. Appl."},{"key":"ref_37","doi-asserted-by":"crossref","first-page":"1355","DOI":"10.1016\/j.fss.2008.11.006","article-title":"An overview of the asymmetric bipolar representation of positive and negative information in possibility theory","volume":"160","author":"Dubois","year":"2009","journal-title":"Fuzzy Sets Syst."},{"key":"ref_38","doi-asserted-by":"crossref","first-page":"3","DOI":"10.1016\/j.fss.2010.11.007","article-title":"Gradualness, uncertainty and bipolarity: Making sense of fuzzy sets","volume":"192","author":"Dubois","year":"2012","journal-title":"Fuzzy Sets Syst."},{"key":"ref_39","doi-asserted-by":"crossref","first-page":"1747","DOI":"10.1186\/s40064-016-3420-x","article-title":"Pricing for a basket of LCDS under fuzzy environments","volume":"5","author":"Wu","year":"2016","journal-title":"SpringerPlus"},{"key":"ref_40","doi-asserted-by":"crossref","first-page":"340","DOI":"10.1080\/18756891.2016.1161345","article-title":"A New Default Intensity Model with Fuzziness and Hesitation","volume":"9","author":"Wu","year":"2016","journal-title":"Int. J. Comput. Intell. Syst."},{"key":"ref_41","doi-asserted-by":"crossref","first-page":"3481863","DOI":"10.1155\/2018\/3481863","article-title":"A New Default Probability Calculation Formula an Its Application under Uncertain Environments","volume":"2018","author":"Wu","year":"2018","journal-title":"Discret. Dyn. Nat. Soc."},{"key":"ref_42","doi-asserted-by":"crossref","first-page":"140","DOI":"10.5755\/j01.ee.29.2.19206","article-title":"Intuitionistic fuzzy real-options theory and its application to solar energy investment projects","volume":"29","author":"Ersen","year":"2018","journal-title":"Eng. Econ."},{"key":"ref_43","doi-asserted-by":"crossref","first-page":"2584","DOI":"10.1109\/TEM.2022.3153960","article-title":"Solar Energy Investment Valuation with Intuitionistic Fuzzy Trinomial Lattice Real Option Model","volume":"70","author":"Ersen","year":"2023","journal-title":"IEEE Trans. Eng. Manag."},{"key":"ref_44","doi-asserted-by":"crossref","first-page":"273","DOI":"10.1023\/B:REOM.0000032115.22510.b5","article-title":"Probability\u2013possibility transformations, triangular fuzzy sets, and probabilistic inequalities","volume":"10","author":"Dubois","year":"2004","journal-title":"Reliab. Comput."},{"key":"ref_45","doi-asserted-by":"crossref","unstructured":"Chrysafis, K.A., and Papadopoulos, B.K. (2021). Decision Making for Project Appraisal in Uncertain Environments: A Fuzzy-Possibilistic Approach of the Expanded NPV Method. Symmetry, 13.","DOI":"10.3390\/sym13010027"},{"key":"ref_46","doi-asserted-by":"crossref","unstructured":"Andr\u00e9s-S\u00e1nchez, J. (2023). A Fuzzy-Random Extension of Jamshidian\u2019s Bond Option Pricing Model and Compatible One-Factor Term Structure Models. Axioms, 12.","DOI":"10.3390\/axioms12070668"},{"key":"ref_47","doi-asserted-by":"crossref","first-page":"237","DOI":"10.2307\/2331065","article-title":"The use of the control variate technique in option pricing","volume":"23","author":"Hull","year":"1988","journal-title":"J. Financ. Quant. Anal."},{"key":"ref_48","doi-asserted-by":"crossref","first-page":"338","DOI":"10.1016\/S0019-9958(65)90241-X","article-title":"Fuzzy Sets","volume":"8","author":"Zadeh","year":"1965","journal-title":"Inf. Control"},{"key":"ref_49","doi-asserted-by":"crossref","unstructured":"Dubois, D., Prade, H., and Yager, R.R. (1993). Fuzzy numbers: An overview. Readings in Fuzzy Sets and Intelligent Systems, Elsevier.","DOI":"10.1016\/B978-1-4832-1450-4.50015-8"},{"key":"ref_50","doi-asserted-by":"crossref","first-page":"87","DOI":"10.1016\/S0165-0114(86)80034-3","article-title":"Intuitionistic fuzzy sets","volume":"20","author":"Atanassov","year":"1986","journal-title":"Fuzzy Sets Syst."},{"key":"ref_51","doi-asserted-by":"crossref","first-page":"1151","DOI":"10.3233\/IFS-151722","article-title":"Engineering economic analyses using intuitionistic and hesitant fuzzy sets","volume":"29","author":"Kahraman","year":"2015","journal-title":"J. Intell. Fuzzy Syst."},{"key":"ref_52","doi-asserted-by":"crossref","first-page":"377","DOI":"10.1142\/S0218488504002886","article-title":"Ranking-intuitionistic fuzzy numbers","volume":"12","author":"Mitchell","year":"2004","journal-title":"Int. J. Uncertain. Fuzziness Knowl.-Based Syst."},{"key":"ref_53","doi-asserted-by":"crossref","first-page":"1142","DOI":"10.1109\/TFUZZ.2014.2346246","article-title":"Least-Squares Regression Based on Atanassov\u2019s Intuitionistic Fuzzy Inputs\u2013Outputs and Atanassov\u2019s Intuitionistic Fuzzy Parameters","volume":"23","author":"Arefi","year":"2015","journal-title":"IEEE Trans. Fuzzy Syst."},{"key":"ref_54","first-page":"15","article-title":"A new approach for ranking of intuitionistic fuzzy numbers","volume":"1","author":"Mohan","year":"2020","journal-title":"J. Fuzzy Ext. Appl."},{"key":"ref_55","doi-asserted-by":"crossref","first-page":"249","DOI":"10.1142\/S0218488501000788","article-title":"The necessity of the strong \u03b1-cuts of a fuzzy set","volume":"9","author":"Couso","year":"2001","journal-title":"Int. J. Uncertain. Fuzziness Knowl. Based Syst."},{"key":"ref_56","doi-asserted-by":"crossref","first-page":"165","DOI":"10.1016\/S0263-2241(00)00036-1","article-title":"A fuzzy approach for the expression of uncertainty in measurement","volume":"29","author":"Mauris","year":"2001","journal-title":"Measurement"},{"key":"ref_57","doi-asserted-by":"crossref","first-page":"512","DOI":"10.1007\/s00500-004-0368-5","article-title":"Fuzzy statistics: Hypothesis testing","volume":"9","author":"Buckley","year":"2005","journal-title":"Soft Comput."},{"key":"ref_58","doi-asserted-by":"crossref","first-page":"345","DOI":"10.1007\/s00500-010-0619-6","article-title":"On Buckley\u2019s approach to fuzzy estimation","volume":"15","author":"Falsafain","year":"2011","journal-title":"Soft Comput."},{"key":"ref_59","doi-asserted-by":"crossref","first-page":"446","DOI":"10.1016\/j.ins.2014.03.131","article-title":"Nonasymptotic fuzzy estimators based on confidence intervals","volume":"279","author":"Sfiris","year":"2014","journal-title":"Inf. Sci."},{"key":"ref_60","doi-asserted-by":"crossref","first-page":"201","DOI":"10.1007\/s12530-019-09285-6","article-title":"Fuzzy-statistical prediction intervals from crisp regression models","volume":"11","author":"Adjenughwure","year":"2020","journal-title":"Evol. Syst."},{"key":"ref_61","doi-asserted-by":"crossref","unstructured":"Al-Kandari, M., Adjenughwure, K., and Papadopoulos, K. (2020). A Fuzzy-Statistical Tolerance Interval from Residuals of Crisp Linear Regression Models. Mathematics, 8.","DOI":"10.3390\/math8091422"},{"key":"ref_62","doi-asserted-by":"crossref","first-page":"1","DOI":"10.3233\/WEB-170349","article-title":"Directional prediction of stock prices using breaking news on Twitter","volume":"15","author":"Alostad","year":"2017","journal-title":"Web Intell."},{"key":"ref_63","doi-asserted-by":"crossref","first-page":"215","DOI":"10.1007\/s10700-012-9150-9","article-title":"Intuitionistic fuzzy linear regression analysis","volume":"12","author":"Parvathi","year":"2013","journal-title":"Fuzzy Optim. Decis. Mak."},{"key":"ref_64","doi-asserted-by":"crossref","first-page":"309","DOI":"10.1016\/0165-0114(90)90204-J","article-title":"On using \u03b1-cuts to evaluate fuzzy equations","volume":"38","author":"Buckley","year":"1990","journal-title":"Fuzzy Sets Syst."},{"key":"ref_65","doi-asserted-by":"crossref","first-page":"167","DOI":"10.1016\/0304-405X(76)90024-6","article-title":"The pricing of commodity contracts","volume":"3","author":"Black","year":"1976","journal-title":"J. Financ. Econ."},{"key":"ref_66","doi-asserted-by":"crossref","first-page":"831","DOI":"10.1016\/j.camwa.2007.01.011","article-title":"Option valuation model with adaptive fuzzy numbers","volume":"53","author":"Thiagarajah","year":"2007","journal-title":"Comput. Math. Appl."},{"key":"ref_67","doi-asserted-by":"crossref","first-page":"1624","DOI":"10.1007\/s40815-018-0468-5","article-title":"Pricing European Options with Triangular Fuzzy Parameters: Assessing Alternative Triangular Approximations in the Spanish Stock Option Market","volume":"20","year":"2018","journal-title":"Int. J. Fuzzy Syst."},{"key":"ref_68","doi-asserted-by":"crossref","first-page":"1025","DOI":"10.1007\/s40815-017-0308-z","article-title":"Value Function Computation in Fuzzy Models by Differential Evolution","volume":"19","author":"Guerra","year":"2017","journal-title":"Int. J. Fuzzy Syst."},{"key":"ref_69","doi-asserted-by":"crossref","first-page":"14","DOI":"10.1016\/j.fss.2016.12.005","article-title":"The application of nonlinear fuzzy parameters PDE method in pricing and hedging European options","volume":"331","author":"Li","year":"2018","journal-title":"Fuzzy Sets Syst."},{"key":"ref_70","doi-asserted-by":"crossref","first-page":"13351","DOI":"10.1007\/s00500-019-03876-w","article-title":"Option pricing and the Greeks under Gaussian fuzzy environments","volume":"23","author":"Chen","year":"2019","journal-title":"Soft Comput."},{"key":"ref_71","doi-asserted-by":"crossref","first-page":"59","DOI":"10.1016\/j.fss.2013.01.010","article-title":"On an implicit assessment of fuzzy volatility in the Black and Scholes environment","volume":"223","author":"Capotorti","year":"2013","journal-title":"Fuzzy Sets Syst."},{"key":"ref_72","doi-asserted-by":"crossref","first-page":"391","DOI":"10.3166\/jds.12.391-416","article-title":"Fuzzy Black and Scholes real options pricing","volume":"12","author":"Collan","year":"2003","journal-title":"J. Decis. Syst."},{"key":"ref_73","first-page":"135","article-title":"A Comparative Fuzzy Real Options Valuation Model using Trinomial Lattice and Black-Scholes Approaches: A Call Center Application","volume":"16","author":"Tolga","year":"2010","journal-title":"J. Mult.-Valued Log. Soft Comput."},{"key":"ref_74","first-page":"693","article-title":"Interval-valued and circular intuitionistic fuzzy present worth analyses","volume":"33","author":"Kahraman","year":"2022","journal-title":"Informatica"},{"key":"ref_75","doi-asserted-by":"crossref","first-page":"108757","DOI":"10.1016\/j.ijpe.2022.108757","article-title":"Intuitionistic fuzzy risk adjusted discount rate and certainty equivalent methods for risky projects","volume":"257","author":"Kahraman","year":"2023","journal-title":"Int. J. Prod. Econ."},{"key":"ref_76","doi-asserted-by":"crossref","unstructured":"Uzhga-Rebrov, O., and Grabusts, P. (2023). Methodology for Environmental Risk Analysis Based on Intuitionistic Fuzzy Values. Risks, 11.","DOI":"10.3390\/risks11050088"},{"key":"ref_77","doi-asserted-by":"crossref","unstructured":"Andr\u00e9s-S\u00e1nchez, J.D. (2024). Pricing Life Contingencies Linked to Impaired Life Expectancies Using Intuitionistic Fuzzy Parameters. Risks, 12.","DOI":"10.3390\/risks12020029"},{"key":"ref_78","first-page":"753852","article-title":"Markov Switching Model Analysis of Implied Volatility for Market Indexes with Applications to S&P 500 and DAX","volume":"2014","author":"Vettori","year":"2014","journal-title":"J. Math."},{"key":"ref_79","first-page":"444","article-title":"Multitask machine learning for financial forecasting","volume":"12","author":"Honchar","year":"2018","journal-title":"Int. J. Circuits Syst. Signal Process."}],"container-title":["Axioms"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/www.mdpi.com\/2075-1680\/13\/8\/503\/pdf","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2025,10,10]],"date-time":"2025-10-10T15:24:31Z","timestamp":1760109871000},"score":1,"resource":{"primary":{"URL":"https:\/\/www.mdpi.com\/2075-1680\/13\/8\/503"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2024,7,26]]},"references-count":79,"journal-issue":{"issue":"8","published-online":{"date-parts":[[2024,8]]}},"alternative-id":["axioms13080503"],"URL":"https:\/\/doi.org\/10.3390\/axioms13080503","relation":{},"ISSN":["2075-1680"],"issn-type":[{"value":"2075-1680","type":"electronic"}],"subject":[],"published":{"date-parts":[[2024,7,26]]}}}