Front fixing implicit finite difference for the American put options model

Abstract

This thesis proposes a unified analytical-numerical framework that addresses two interconnected challenges in modern mathematics:variable-order fractional stochastic differential equations (FSDEs) and the time-fractional Black-Scholes (tfBS) equation for American put options (APOs). For APOs, this thesis develops a hybrid methodology that integrates a front-fixing transformation with a Crank-Nicolson finite-difference scheme, augmented by the Caputo-Fabrizio fractional derivative to capture long-range dependence in financial markets. Financial indices such as the FTSE 100 exhibit persistent autocorrelations with power-law decay, while emerging market indices such as the Bovespa (Brazil) display pronounced long-memory behavior, motivating the use of fractional option pricing models. The proposed approach transforms the free-boundary problem arising from early exercise into a fixed-domain formulation to avoid numerical complexities, enhancing computational stability and efficiency. The resulting numerical scheme is consistent O(1) + O((Δy)2), unconditionally stable and second-order convergent in space, while preserving positivity and monotonicity. Numerical experiments on a benchmark problem (r = 0.1, σ = 0.2, T = 1, α = 0.5) confirm second-order convergence as the spatial mesh is refined. A direct comparison with the existing first-order scheme [163] demonstrates that the proposed method achieves approximately half the error for the same grid resolution. We address a class of non-Lipschitz FSDEs governed by variable-order fractional operators with non-singular kernels. The absence of a convolution structure due to the variable-order fractional calculus necessitates novel analytical techniques. The well-posedness of the solution is established, moment estimates are derived, and a generalised Euler-Maruyama (EM) scheme is constructed. Strong convergence of the proposed method is proved under non-standard regularity conditions, together with explicit mean-square error estimates. The proposed framework reduces to classical stochastic calculus in the limiting regimes (α → 0 and α → 1), thereby recovering the well-established convergence rates of SDEs. Numerical experiments demonstrate the robustness, accuracy, and stability of the proposed scheme in the presence of multiplicative noise and dynamic variables of variable order. Collectively, this study establishes a unified analytical and numerical framework for Caputo- Fabrizio fractional models by integrating fractional stochastic calculus with computational finance. The numerical analysis developed for variable-order FSDEs provides the theoretical foundation for the financial application. In turn, the APO pricing problem demonstrates the practical relevance of fractional calculus (FC) in modelling complex financial systems with memory effects. This approach bridges general stochastic systems with applications in derivative pricing, risk management, and interdisciplinary applied mathematics, while offering robust and scalable methodologies for the analysis of path-dependent phenomena and highdimensional stochastic models.