Random Neural Networks for Rough Volatility

Area of Science:

• Quantitative Finance
• Computational Mathematics
• Machine Learning

Background:

• Path-dependent partial differential equations (PDEs) are crucial in financial modeling, especially for rough volatility.
• Solving these complex equations analytically is often intractable.
• Existing numerical methods may face challenges with high dimensionality and rough volatility dynamics.

Purpose of the Study:

• To develop a novel deep learning-based numerical algorithm for solving path-dependent PDEs in rough volatility modeling.
• To leverage recent advancements in stochastic differential equations and neural network architectures.
• To provide a theoretically grounded and computationally efficient solution.

Main Methods:

• Interpreting the partial differential equation (PDE) as a solution to a backward stochastic differential equation (BSDE).
• Utilizing a reservoir-type neural network architecture, inspired by Gonon, Grigoryeva, and Ortega.
• Formulating the optimization problem as a simple least-squares regression.

Main Results:

• The proposed deep learning algorithm effectively solves path-dependent PDEs relevant to rough volatility.
• The reservoir neural network approach simplifies the optimization to a least-squares regression problem.
• Theoretical convergence properties of the reservoir approach for this problem are established.

Conclusions:

• Deep learning, specifically reservoir neural networks, offers a powerful tool for tackling complex financial PDEs.
• The BSDE interpretation combined with reservoir networks provides a convergent and efficient numerical method.
• This approach advances the computational techniques available for rough volatility modeling.