In traditional finance, option prices are typically calculated using crisp sets of variables. However, as reported in the literature novel, these parameters possess a degree of fuzziness or uncertainty. This allows participants to estimate option prices based on their risk preferences and beliefs, considering a range of possible values for the parameters. This paper presents a comprehensive review of existing work on fuzzy fractional Brownian motion and proposes an extension in the context of financial option pricing. In this paper, we define a unified framework combining fractional Brownian motion with fuzzy processes, creating a joint product measure space that captures both randomness and fuzziness. The approach allows for the consideration of individual risk preferences and beliefs about parameter uncertainties. By extending Merton’s jump-diffusion model to include fuzzy fractional Brownian motion, this paper addresses the modelling needs of hybrid systems with uncertain variables. The proposed model, which includes fuzzy Poisson processes and fuzzy volatility, demonstrates advantageous properties such as long-range dependence and self-similarity, providing a robust tool for modelling financial markets. By incorporating fuzzy numbers and the belief degree, this approach provides a more flexible framework for practitioners to make their investment decisions.

In traditional finance, option prices are typically calculated using crisp sets of variables. However, in a proposed new approach, the assumption is made that these parameters have a degree of fuzziness or uncertainty associated with them. This allows participants to estimate option prices based on their risk preferences and beliefs, considering a range of possible values for the parameters. To determine the belief degree associated with a particular option price, an interpolation search algorithm was proposed recently[1]. Interpolation is a mathematical technique used to estimate values between known data points. In this case, the algorithm is applied to determine the belief degree associated with a specific option price within the range of possible prices. By incorporating fuzzy numbers and the belief degree, this approach provides a more flexible framework for practitioners to make their investment decisions. It allows them to consider their risk preferences and beliefs about the uncertain parameters when selecting an option price. In this paper, we will review a unified framework for combining fractional Brownian motion with fuzzy processes and obtain a joint measure space. We start by constructing a product measure space that captures the randomness and fuzziness inherent in the respective processes. We begin with the usual Kolmogorov filtered probability space (Ω, F, ℙ), where ℙ represents the probability measure, Ω is the set of all possible outcomes or states that a random process can take and {Ft, t ∈ [0, T]} is an information filtration. We will review the work of Merton (1976) on option pricing when the underlying stock returns have jumps [2] and expand it to fuzzy fractional processes.

Clustering involves gathering a collection of objects into homogeneous groups or clusters, such that objects in the same cluster are more similar when compared to objects present in other groups. Clustering algorithms that generate a tree of clusters called dendrogram which can be either divisive or agglomerative. The partitional clustering gives a single partition of objects, with a predefined K number of clusters. The most popular partition clustering approaches are: k-means and fuzzy C-means (FCM). In k-means clustering, data are divided into a number of clusters where data elements belong to exactly one cluster. The k-means clustering works well when data elements are well separable. To overcome the problem of non-separability, FCM and IFCM clustering algorithm were proposed. Here we review the use of FCM/IFCM with reference to the problem of market volatility.

Volatility moves in financial markets are rare and sporadic i.e., periods of low volatility typically follow each other until the regime change due to technical and/or fundamental exogenous factors. Subsequently higher volatility tends to lead to even higher volatility and vice versa. Such dynamic systems require alternative ways of representation than a static multi- layer perceptron (MLP). The prevalent models assume L-stable distributions with independent and stable increments. This contrasts with what is observed in real world. L-stable distributions miss one of the main features of financial markets - the alternation of periods of large price changes with periods of small price changes. To correct for this deficiency another, self-affine process had been introduced - Fractional Brownian Motion (FBM). FBM does not capture fat tails or fluctuations in volatility that are unrelated to the predictability of future returns. In summary, both models have strong scale-invariance property, in which the distribution of returns over different sampling intervals are identical. This property is clearly at odds with empirical observations.The key idea in this paper is that Generative Adversarial Networks (GAN) combined with fractality of data is better suited to manage volatility time series between different shock events because it is structured to maintain a memory of older points in the time series and continuously learn from them. Episodic memory that is used in GANs maintains explicit record of past events. In order to make decision, the action is chosen that has the highest value based on the outcomes of past similar situations. There are other, older models such as time-delayed window input vector (TDNN nets) and recurrent structures such as ‘Jordan’ and ‘Elman’ but in this study we focus on a more modern way of looking at dynamic time series through semi- supervised framework.

In this study, we present a novel mathematical framework for pricing financial derivates and modelling asset behaviour by bringing together fractional Brownian motion (fBm), fuzzy logic, and jump processes, all aligned with no-arbitrage principle. In particular, our mathematical developments include fBm defined through Mandelbrot-Van Ness kernels, and advanced mathematical tools such Molchan martingale and BDG inequalities ensuring rigorous theoretical validity. We bring together these different concepts to model uncertainties like sudden market shocks and investor sentiment, providing a fresh perspective in financial mathematics and derivatives pricing. By using fuzzy logic, we incorporate subject factors such as market optimism or pessimism, adjusting volatility dynamically according to the current market environment. Fractal mathematics with the Hurst exponent close to zero reflecting rough market conditions and fuzzy set theory are combined with jumps, representing sudden market changes to capture more realistic asset price movements. We also bridge the gap between complex stochastic equations and solvable differential equations using tools like Feynman-Kac approach and Girsanov transformation. We present simulations illustrating plausible scenarios ranging from pessimistic to optimistic to demonstrate how this model can behave in practice, highlighting potential advantages over classical models like the Merton jump diffusion and Black-Scholes. Overall, our proposed model represents an advancement in mathematical finance by integrating fractional stochastic processes with fuzzy set theory, thus revealing new perspectives on derivative pricing and risk-free valuation in uncertain environments.

We present the first application of the Multifractal Model of Asset Returns (MMAR) to an implied volatility index, using 36 years of daily CBOE VIX observations spanning four economic cycles. Three general conclusions emerge. First, implied volatility is multifractal: its scaling function is strictly concave, and this curvature survives explicit comparison against monofractal, ARMA, and ARFIMA nulls fitted to the same data, so it cannot be reproduced by anti-persistence or short-range linear dependence alone. Second, unlike equity price indices which are persistent, the VIX is strongly subdiffusive ($\widehat{H}\approx 0.18$, far below $\frac{1}{2}$), which is the multifractal signature of its mean-reverting character; the lognormal cascade is nonetheless admissible, so the construction is internally consistent. Third, admissibility notwithstanding, the lognormal cascade is insufficient in the extreme tails. Across Monte Carlo validation, higher-moment and tail-risk (VaR/ES) comparisons, and a GARCH/EGARCH/FIGARCH benchmark, it captures the bulk of the distribution but systematically underestimates the most violent volatility spikes and does not reproduce VIX’s pronounced positive skewness. We quantify this: the admissible cascade recovers about 84% of the excess kurtosis and reproduces 95–99% Value-at-Risk and 95% Expected Shortfall almost exactly, but it understates the deepest Expected Shortfall, and, being symmetric, it cannot reproduce the positive skew, underpricing far-out-of-the-money option premia by up to 100%. The indicated direction is asymmetric, heavier-tailed cascade extensions. Beyond VIX, the analysis offers a reproducible template for distinguishing genuine multifractality from its linear imitators in any volatility series.