Generalizing subdiffusive Black-Scholes model by variable exponent: Model transformation and numerical approximation
Meihui Zhang, Yaxue Liu, Mengmeng Liu, Wenlin Qiu, Xiangcheng Zheng
TL;DR
The paper addresses pricing under a subdiffusive Black-Scholes model with a time-varying memory exponent $0<\alpha(t)<1$, which introduces a challenging variable-exponent memory kernel. It combines a nonlinear-to-linear transformation with a spatial-temporal transform and a convolution-based reformulation to obtain a tractable, coercive PDE suitable for numerical discretization. The authors develop a time-discrete and a fully discrete finite element scheme, prove stability and convergence with a temporal rate $\tau^{\tfrac{1}{2}+\tfrac{3}{2}\alpha_0}$ and a spatial rate $h^2$, and corroborate the theory with numerical experiments. The work advances option pricing under memory effects and lays groundwork for extensions to more general volatility structures and stochastic dynamics.
Abstract
This work generalizes the subdiffusive Black-Scholes model by introducing the variable exponent in order to provide adequate descriptions for the option pricing, where the variable exponent may account for the variation of the memory property. In addition to standard nonlinear-to-linear transformation, we apply a further spatial-temporal transformation to convert the model to a more tractable form in order to circumvent the difficulties caused by the ``non-positive, non-monotonic'' variable-exponent memory kernel. An interesting phenomenon is that the spatial transformation not only eliminates the advection term but naturally turns the original noncoercive spatial operator into a coercive one due to the specific structure of the Black-Scholes model, which thus avoids imposing constraints on coefficients. Then we perform numerical analysis for both the semi-discrete and fully discrete schemes to support numerical simulation. Numerical experiments are carried out to substantiate the theoretical results.