Paper: arXiv 2602.00201

Authors: Neetu Garg, A. S. V. Ravi Kanth

Abstract

This paper implements an efficient numerical algorithm for the time-fractional Black-Scholes model governing European options. The proposed method comprises the Crank-Nicolson approach to discretize the time variable and exponential B-spline approximation for the space variable. The implemented method is unconditionally stable. We present few numerical examples to confirm the theory. Numerical simulations with comparisons exhibit the supremacy of the proposed approach.

Complexity vs Empirical Score

  • Math Complexity: 6.5/10
  • Empirical Rigor: 2.0/10
  • Quadrant: Lab Rats — theoretically deep, empirically untested

Why this score: The paper presents a solid mathematical formulation using Caputo fractional derivatives, Crank-Nicolson discretization, and exponential B-spline approximations. However, it lacks empirical financial rigor, testing only on a toy mathematical problem with a known exact solution rather than real-world market option data or backtests.

Research Flowchart

  flowchart TD
    A[Goal: Efficiently price European options using Time-Fractional Black-Scholes Model] --> B[Inputs: Option Parameters & Financial Market Data]
    B --> C[Time Discretization: Crank-Nicolson Scheme]
    C --> D[Space Discretization: Exponential B-Spline Approximation]
    D --> E[Computational Process: Solve Discrete System & Prove Unconditional Stability]
    E --> F[Validation: Numerical Simulations & Comparative Analysis]
    F --> G[Outcome: Stable, highly accurate option pricing algorithm superior to existing methods]

Cited References (18)

  1. Conformal Finite Element Methods for Nonlinear Rosenau-Burgers-Biharmonic Models (2024) — link
  2. A new error estimates of finite element method for (2+1)-dimensional nonlinear advection-diffusion model (2023) — link
  3. New multiple analytic solitonary solutions and simulation of (2+1)-dimensional generalized Benjamin-Bona-Mahony-Burgers model (2023) — link
  4. Numerical Solution of Time Fractional Black–Scholes Model Based on Legendre Wavelet Neural Network with Extreme Learning Machine (2022) — link
  5. A computational procedure and analysis for multi‐term time‐fractional Burgers‐type equation (2022) — link
  6. An unconditionally stable algorithm for multiterm time fractional advection–diffusion equation with variable coefficients and convergence analysis (2020) — link
  7. A high accuracy numerical method and its convergence for time-fractional Black-Scholes equation governing European options (2020) — link
  8. A Computational Method Based on the Moving Least-Squares Approach for Pricing Double Barrier Options in a Time-Fractional Black–Scholes Model (2019) — link
and 10 more
  1. Selection of shape parameter in radial basis functions for solution of time-fractional Black-Scholes models (2018) — link
  2. Numerically pricing double barrier options in a time-fractional Black-Scholes model (2017) — link
  3. Numerical solution of the time fractional Black-Scholes model governing European options (2016) — link
  4. Theory of Rational Option Pricing (2015) — link
  5. Solution of the Fractional Black-Scholes Option Pricing Model by Finite Difference Method (2013) — link
  6. Theory of exponential splines (1991) — link
  7. The Pricing of Options and Corporate Liabilities (1973) — link
  8. A new difference scheme for time fractional heat equations based on the Crank-Nicolson method (2013)
  9. THE FRACTIONAL BLACK-SCHOLES EQUATION (2000)
  10. Analysis and simulation of Korteweg-de Vries-Rosenau-regularised long-wave model via Galerkin finite element method

0 of 18 cited works are in the corpus; resolved via arXiv source bundles, Semantic Scholar and OpenAlex. Corpus-wide: 2518 linked papers.