We present a neural network approach for calibrating surfaces for an option price and an implied volatility satisfying sufficient conditions for no arbitrage by using the physics-informed neural network (PINN). We find an approximate solution for option price and implied volatility of the Black-Scholes-Merton dual equation, which has exactly the same form as the Black-Scholes-Merton equation. A solution of the BlackScholes-Merton dual equation guarantees arbitrage-free because it was derived from the Black-Scholes-Merton equation. We introduce a neural network for option price and train it by minimizing the losses of the Black-Scholes-Merton dual equation, initial condition, and boundary condition. Furthermore, we additionally introduce and train a neural network for implied volatility by minimizing the loss of market implied volatility. This approach enables us to estimate option price and implied volatility simultaneously, which satisfy sufficient conditions for no arbitrage and approximate market price and market implied volatility. Through the optimization process, neural networks are trained to reproduce an observed market data and predict new option price and implied volatility. Our method describes a surface of option price and implied volatility on the entire domain by a unique function and easily obtain a local volatility by an automatic differentiation.
