Numerical Simulations for Time-Fractional Black Scholes Equations

Battle Count: 90

The paper addresses option pricing, a core component of quantitative trading. The time-fractional Black-Scholes model captures memory effects and jumps in asset prices, which are relevant for derivatives pricing and hedging. However, the paper is primarily a numerical methods contribution rather than a trading strategy paper. Practical relevance depends on whether fractional models provide better pricing than classical Black-Scholes for real market data, which is not demonstrated here.

The method requires: (1) understanding of fractional calculus (Caputo derivative discretization), (2) Crank-Nicolson time-stepping with history terms, (3) exponential B-spline basis function construction with parameter p, (4) assembly of tridiagonal systems at each time step, (5) boundary condition handling. The history-dependent terms in the fractional derivative add computational complexity (O(N^2) memory/time per step). Moderate difficulty for someone familiar with numerical PDE methods.

The paper provides detailed mathematical formulations, discretization schemes, parameter choices (r=0.05, sigma=0.25, D=0, T=1), and test problems with known exact solutions. However, no code or repository is provided. The methodology is fully described mathematically, allowing reimplementation, but practical implementation details (e.g., choice of parameter p, solver for tridiagonal system) are somewhat implicit.