We develop and analyse a barrier-aligned Chebyshev integral tau method for pricing double-barrier and rebate options under the Black–Scholes model. The two barriers define the computational corridor exactly, and the interior strike kink of the payoff is placed on an element interface; both are enforced through finite-rank tau rows rather than through interpolation. On each element the unknown is the Chebyshev coefficient vector of the second derivative together with two integration constants, so the spatial operator is a banded block plus a low-rank border. We give the construction explicitly and provide a complete analysis: a detailed proof of elementwise analytic coefficient decay; a stability theory comprising the sectoriality (dissipativity) of the discrete generator, the A- and L-stability of the time integrators, and a uniform-in-degree bound on the spectral radius of the discrete propagator; a semidiscrete spectral error bound; and a fully discrete second-order bound under Rannacher–Crank–Nicolson time stepping. The implicit systems are solved by a Sherman–Morrison–Woodbury elimination of the interface constants in O(s+r3+rs) work, with s the number of coefficients and r the number of interface constraints. Every result is tested against an exact Dirichlet eigenfunction benchmark, itself validated by Monte Carlo. Strike alignment lowers the payoff-projection error from 1.7×10−1 to 1.3×10−13 and the priced error from ∼10−3 to the temporal floor; Rannacher startup restores second-order time accuracy (1.00→2.03); deltas and gammas are recovered to ∼10−9; the spectrum of the discrete generator is shown to lie on the negative real axis; the propagator is uniformly stable in the degree; and the bordered solver is up to 26× faster than a dense factorisation at the sizes tested.
Standard financial models often fail to capture the intricate, non-linear dynamics of real-world markets. We develop a symmetry-based analysis of an option-pricing model that exhibits long-memory effects, obtained by driving the underlying with a fractional Brownian motion. We focus on the pricing equation for a European-exercise, fixed-strike geometric-average (Asian) option and carry out a complete Lie point-symmetry classification of that equation in logarithmic coordinates, in which the path-dependent averaging term takes a tractable form. The classification yields a solvable six-dimensional symmetry algebra, several of whose generators are genuinely fractional, carrying weights t2H,t2H+1 tied to the Hurst parameter, together with the usual infinite-dimensional superposition freedom of a linear equation. Because the pricing operator is parabolic and is not the Euler–Lagrange equation of a variational principle, Noether’s theorem does not apply directly; instead we construct conservation laws through Ibragimov’s nonlinear self-adjointness (equivalently, through the direct multiplier method) by introducing the adjoint pricing operator. The resulting conserved currents are parametrised by solutions of the adjoint (state-price) equation and encode the martingale/no-arbitrage structure of the model. A discount-weighted adjoint solution yields an explicit, exactly verifiable conservation law, and pairing the adjoint solutions with the symmetry generators through Ibragimov’s theorem produces a bilinear, symmetry-indexed family of such laws, three of which are genuinely fractional. Numerical experiments analyse the effect of the Hurst parameter on prices and Greeks, verify the conservation law against the discretisation error, and compare the fractional model with the standard Black–Scholes benchmark.
Financial markets, particularly those involving risky assets such as stocks, corporate bonds, and derivatives, play a crucial role in economic growth, investment, and wealth distribution. Understanding the dynamics of these markets poses a significant challenge in financial mathematics due to the intricate interplay of delay effects and nonlinear behavior. This study simulates a two-delay pricing model in a risky financial market, formulated as an integro-differential equation containing two distinct hyperbolic terms. To improve modeling accuracy, the equation is reformulated as a fractional-order integro-differential equation, since fractional derivatives provide greater flexibility in capturing the memory-dependent behavior of complex systems. For the numerical solution, a discrete collocation method based on local radial basis functions (LRBFs) is employed. Its localized nature substantially reduces memory requirements and computational costs by solving several smaller systems instead of a single large one. Moreover, the algorithm is straightforward to implement and compatible with standard personal computers. To evaluate the effectiveness of the offered approach, we conduct an error analysis and present several numerical examples. The results demonstrate that the method offers improved efficiency compared to its global counterpart and align well with theoretical predictions, confirming both the accuracy and convergence of the scheme.
The problem of finding the extremal function of an interpolation formula and of evaluating the norm of the error functional in the Sobolev space was first posed by S.L.Sobolev. In that work (Sobolev, 1961 [1]), the extremal function of the interpolation formula was obtained explicitly in the Sobolev space whose generalized derivatives of order m are square-integrable. The present paper considers a new optimal interpolation formula in the Sobolev space: using the discrete analog of the differential operator, we obtain the optimal coefficients of the interpolation formula for an arbitrary m, and the norm of the error functional is computed. Integration of the interpolant leads to a quadrature formula; three numerical examples confirm the theoretical order of convergence.
This manuscript studies stochastic differential equations involving the Atangana-Baleanu-Caputo fractional operator. The operator contains a nonsingular Mittag-Leffler memory kernel, and therefore it avoids the strong initial singularity that appears in classical Caputo-type kernels while retaining hereditary effects. The main stochastic equation is reformulated through the Atangana-Baleanu integral in Ito sense. Existence, uniqueness, second-moment boundedness, continuous dependence, Hyers-Ulam stability, generalized Hyers-Ulam stability, and mean-square stability are established under Lipschitz, linear-growth, time-regularity, and compatibility assumptions. A memory finite-difference Euler–Maruyama method is constructed for the corresponding Atangana-Baleanu stochastic integral equation, including a local implicit correction induced by the nonsingular-kernel formulation. Artificial neural-network and deep residual-network surrogates are then introduced for learning-assisted approximation of finite-difference trajectories. Numerical examples, tables, and section-specific figures are provided for the kernel behavior, moment bounds, stability indicator, finite-difference memory weights, sample paths, convergence, training loss, residual error, and surrogate comparison.
