A systematic review of the interactions of fuzzy set theory and option pricing

Fuzzy Random Option Pricing in Continuous Time: A Systematic Review and an Extension of Vasicek’s Equilibrium Model of the Term Structure

“…Alternatively, following [31,32,36,52,77,94], we can adjust the parameters that are assumed to be fuzzy numbers based on existing evidence in financial markets. Likewise, it should be noted that the existence of studies that empirically apply FROPCP developments is relatively scarce [19,20]. Both considerations motivate the parameter adjustment methodology of the mean reversion process outlined in this study.…”

“…However, the strike price and expiration are crisp parameters because they are clearly defined in the contract. However, in real options, the strike price [33] or even the expiration [19] may not be known with precision and, therefore, are susceptible to be quantified with fuzzy numbers.…”

“…We must acknowledge that empirical applications of FROPCP are relatively scarce [19,20]. Within this group of works, we can highlight [31,32], which propose the use of congruent probability-possibility transformations [105] to model the volatility of options based on empirical data, and [71][72][73], which use fuzzy regression models to estimate the volatility, kurtosis, and skewness of asset returns.…”

“…This paper presents a double contribution. On the one hand, it has systematically reviewed a set of contributions that we can label as fuzzy-random option price in continuous time (FROPCT), which is the dominant approach within fuzzy option pricing [19]. On the other hand, we have extended the fuzzy-random approach to model the term structure with an equilibrium model.…”

A Systematic Overview of Fuzzy Random Option Pricing in Discrete Time and a Binomial Extension to Temporal Structure of Interest Rates

“…Within the models in continuous time, we discriminate those developed within the BSM framework, i.e., based on the hypothesis of the geometric Brownian motion of subjacent asset growth (49 contributions), and those based on more complex stochastic modelling. The BSM group includes papers that fuzzify the adaptation of BSM to currency options by Garman and Kolhagen (1983), such as Xu et al (2013), or exchange option pricing by Margrabe (1978) by Anzilli & Villani (2021, 2023. The fuzzy approximation to other random continuous models (27 papers) embeds the jumpdiffusion model (Merton, 1976) by Xu et al (2009); Heston's stochastic variance (Heston, 1993), in (Figà-Talamanca, Guerra & Stefanini, 2012; fractional Brownian motion (Ghasemalipour, Fathi-Vajargah, 2019) and Levy processes (Nowak & Romaniuk, 2013;Nowak, Pawlowski, 2017).…”