In this article, we solve the Black-Scholes Partial Difference Equation(BS PDE) under the local volatility model by using an artificial neural network, and obtain the price and the greeks of derivatives in forms of function simultaneously. Dupire’s local volatility model is one of the most successful for equity models. In practice, it is important to price and hedge derivatives under the local volatility model. We provide an artificial neural network scheme for efficiently solving parametric PDEs.
We adopt local volatility models, especially, constant elasticity of variance(CEV) model and volatility surface. The price function of European vanilla options under the local volatility model satisfies Dupire's equation. We solve the parametric PDE of the European put option under the local volatility model with an artificial neural network and show that solution and Dupire's equation are approximated.
