Neural Networks and the Nonlinear Feynman–Kac Theorem Applied to Financial Options Pricing

The classic financial market model proposed by Black and Scholes allows solving, under a series of simplifying assumptions, the problem of valuing financial derivatives. However, these assumptions deviate from the reality of global markets, particularly assuming perfect liquidity. In this work, a financial market model is proposed in which the illiquidity of the risky asset is stochastic, described by a mean-reversion process. The dynamics of the asset price in this context are established alongside the valuation PDE for derivatives. Additionally, as an approximation method to the solution of this PDE, the implementation of the extension of the Feynman–Kac representation theorem to the nonlinear case using deep neural networks is proposed.