FRACTAL BASED FRAMEWORK FOR TIME SERIES VOLATILITY PREDICTION

In this thesis, a novel mathematical framework is presented for pricing financial derivatives and modelling asset behaviour by bringing together fractional Brownian motion (fBm), fuzzy logic, and jump processes, all aligned with the 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. These different concepts are combined to model uncertainties such as sudden market shocks and investor sentiment, providing a fresh perspective in financial mathematics and derivatives pricing. Using fuzzy logic to 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. The gap between complex stochastic equations and solvable differential equations is bridged using tools like Feynman–Kac approach and Girsanov transformation. Simulations illustrating plausible scenarios ranging from pessimistic to optimistic demonstrate how this model can behave in practice, highlighting potential advantages over classical models like the Merton jump diffusion and Black–Scholes. Overall, the 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.

In financial modelling, traditional approaches often rely on deterministic frameworks that inadequately capture the inherent complexities of market dynamics. For example, option pricing models typically use crisp parameters, yet empirical evidence shows these variables often exhibit fuzziness and uncertainty. To address these limitations, this thesis proposes a novel unified framework that integrates fractional Brownian motion (fBM) and fuzzy processes to model financial systems characterised by both randomness and fuzziness. By defining a joint product measure space for H ∈ (0,1) where H is the Hurst parameter that characterises the degree of self–similarity and persistence in fractional Brownian motion, this work provides a comprehensive framework to account for long–range dependence, self–similarity, and individual risk preferences in market behaviour. Building on the Merton jump–diffusion model, we extend the framework to include fuzzy fractional Brownian motion, fuzzy Poisson processes, and fuzzy volatility. These innovations enable the modelling of hybrid systems under uncertain variables, offering practitioners a flexible and robust tool for understanding and fore casting financial market dynamics and proposing a novel methodology for its application in forecasting. This thesis examines the fractal characteristics of volatility time series. It provides empirical evidence that supports the proposition of self–similarity across different temporal scales and validates the fractal nature of volatility through Hurst exponent estimation. This finding motivates the development of the novel theoretical framework presented herein. The analysis highlights how fractal measures can capture structured patterns amidst apparent chaos, offering insights into intermittent bursts of volatility and periods of persistence.

Further, the thesis distinguishes between expected and unexpected volatility, linking endogenous shocks (e.g., economic changes) to episodic uncertainties and exogenous shocks (e.g., geopolitical events) to aleatoric uncertainties. By incorporating reinforcement learning (RL) and generative adversarial networks (GANs), the proposed framework manages volatility across diverse shock events, maintaining memory of older data points while continuously adapting to new information. Comparisons between traditional econometric models and modern machine learning approaches are used to benchmark current forecasting capabilities, highlighting the limitations that motivate our new theoretical approach. The research culminates in a novel hybrid framework that lays the groundwork for a new class of more advanced forecasting systems.

This research significantly advances the field by bridging theoretical advance ments in fuzzy stochastic processes and fractal analysis with a clear roadmap for practical applications in financial markets. The findings underscore the importance of incorporating uncertainty and complexity in financial modelling, providing a robust foundation for future studies in stochastic hybrid systems.