An Efficient Physics-Informed Neural Network Solution to the Time-Space Fractional Black-Scholes Equation
Researchers have developed a physics-informed neural network (PINN) method to solve the time-space fractional Black-Scholes equation, a complex model that accounts for memory effects and unusual market behavior. Their approach uses a stable matrix form of the Grunwald-Letnikov fractional derivative, overcoming issues with traditional methods. They introduce a new transformation for the fractional operator, validated by accurate simulations of European put options. However, the study does not include real market data, focusing on numerical results.
What it examines
This paper presents a new method using physics-informed neural networks (PINNs) to solve the time-space fractional Black-Scholes equation, which models option pricing with memory and nonlocal effects. The approach avoids traditional mesh-based methods, aiming for accurate and stable solutions in complex financial models.
What it concludes
The PINN method shows high accuracy and stability for pricing European put options, especially as the fractional order approaches one. This technique can be applied to advanced financial modeling, risk management, and quantitative finance, with potential for further research in solving other fractional and stochastic PDEs.
Evidence objects
Researchers unveil a physics-informed neural network (PINN) to solve the time-space fractional Black-Scholes equation, capturing memory effects and anomalous diffusion in financial markets with unprecedented mathematical rigor and computational stability.
key_findings bullet 1 · key_findings · validation V0
The study introduces a stable matrix formulation of the Grunwald-Letnikov fractional derivative within PINN, overcoming mesh and stability issues of traditional methods, and proposes a new transformation for simplifying the fractional operator.
key_findings bullet 2 · key_findings · validation V0
Numerical experiments on European put options show high accuracy, especially as the fractional order nears one ($\alpha \to 1$), but the absence of real market data limits immediate practical impact despite the methods promise.
key_findings bullet 3 · key_findings · validation V0
This paper innovatively applies physics-informed neural networks (PINNs) to the time-space-fractional Black-Scholes equation, incorporating the Grunwald-Letnikov derivative and a novel spatial operator transformation. Avoiding mesh discretization, it delivers robust European put option results. While building on recent advances, its unique computational approach significantly advances option pricing under fractional dynamics.
key_findings bullet 4 · key_findings · validation V0
Raw abstract and provenance
- … spatial anomalous diffusion in financial markets. Starting from … PDEs in quantitative finance, complementing recent … : Section 2 covers mathematical preliminaries; Section …
Source row: 177 · abstract type: snippet