Quantifying asset price volatility with fractional Brownian motion

Анотація

The object of the study is the behavior of stock market volatility in response to sudden shocks and crisis-driven fluctuations, with a specific focus on capturing its complex temporal structure and memory effects. One of the biggest challenges in this domain lies in the inherent stochastic nature of volatility: it evolves irregularly over time, cannot be directly observed, and must be estimated from indirect indicators. Conventional models, particularly those grounded in classical Brownian motion, often fall short in accurately representing such dynamics, as they neglect the long-range dependence – or “market memory” – commonly observed in real financial time series. This oversight can lead to significant errors in volatility estimation, especially during phases of market turbulence such as financial crises or global events. A fractional diffusion framework was used during the study to model asset price dynamics, incorporating a time-dependent and initially unknown volatility function. This approach relies on fractional Brownian motion, whose non-Markovian properties enable the model to effectively account for long-term correlations in market behavior. To estimate the volatility, it is possible to employ statistical tools based on p-variations, which allowed to compute the Hurst index and reconstruct the underlying path of realized volatility with high sensitivity to structural market changes. It is possible to obtain that this method significantly improves the accuracy of volatility tracking, particularly under stress conditions, such as those observed during the 2020 COVID-19 crisis. It is connected to the fact that the suggested method has a number of features, in particular its ability to incorporate memory effects and to respond adaptively to high-frequency data variations. Thanks to that, let’s manage to capture abrupt volatility spikes and sustained market uncertainty more precisely. Compared to the standard models, it is possible to achieve the following advantages: enhanced responsiveness to market dynamics, improved reliability of volatility forecasts during crisis periods, and a more realistic reflection of financial market complexity.

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