Fractal activity time risky asset models with dependence and heavy-tailed distributions

Saidan, Bader 2026. Fractal activity time risky asset models with dependence and heavy-tailed distributions. PhD Thesis, Cardiff University. Item availability restricted.

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Abstract

Since its discovery in 1973, the Black-Scholes model has remained central to quantitative finance. Although the underlying geometric Brownian motion provides a foundational, tractable framework for option pricing, decades of statistical ana lysis have demonstrated that the model fails to capture the stylised facts of re turns. To address these limitations, we consider the fractal activity time geometric Brownian motion proposed by Chris Heyde in 1999. This model accommodates the empirical features observed in the data, such as dependence in absolute and squared returns, but not in the returns themselves, and a marginal distribution with heavier tails and a higher peak than the Gaussian distribution. Specifically, this model generalises geometric Brownian motion by evaluating the standard Brownian motion at a random activity time rather than calendar time. In this thesis, we describe the construction of this fractal activity time via two different methods: Ornstein-Uhlenbeck processes and a superposition of Ornstein Uhlenbeck processes. We then derive new pricing formulas for several options, including digital, power, and barrier options. We calibrate the model to empirical return data using the method of moments and quasi-maximum likelihood estimation. Furthermore, we validate the derived formulae through Monte Carlo simulations. Finally, for the SPY and MSFT option chains considered, the FATGBM model yields lower RMSE values than the classical Black-Scholes benchmark under the calibration used in this study

Item Type:	Thesis (PhD)
Date Type:	Completion
Status:	Unpublished
Schools:	Schools > Mathematics
Subjects:	Q Science > QA Mathematics
Date of First Compliant Deposit:	28 October 2026
Last Modified:	28 Oct 2026 12:08
URI:	https://orca.cardiff.ac.uk/id/eprint/190484

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