Whack-a-mole Learning: Physics-Informed Deep Calibration for Implied Volatility Surface

Calibrating the Implied Volatility Surface (IVS) using sparse market data is an essential task for option pricing in quantitative finance. The calibrated values must provide a solution to a specified partial differential equation (PDE) in addition to obeying no-arbitrage conditions modelled by individual differential inequalities. However, this leads to a multi-objective optimization problem, which emerges in Physics-Informed Neural Networks (PINNs) as well as in our generalized framework. In order to address this problem, we propose a novel calibration algorithm called Whack-a-mole Learning (WamL), which integrates self-adaptive and auto-balancing processes for each loss term. The developed algorithm realizes efficient reweighting mechanisms for each objective function, ensuring alignment with constraints of price derivatives to achieve smooth surface fitting while satisfying PDE and no-arbitrage conditions. In our tests, this approach enables the straightforward implementation of a deep calibration method that incorporates no-arbitrage constraints, providing an appropriate fit for uneven and sparse market data. WamL also enhances the representation of risk profiles for option prices, offering a robust and efficient solution for IVS calibration.