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Volatility smile and skewness are two key properties of option prices thatare represented by the implied volatility (IV) surface. However, IV surfacecalibration through nonlinear interpolation is a complex problem due to severalfactors, including limited input data, low liquidity, and noise. Additionally,the calibrated surface must obey the fundamental financial principle of theabsence of arbitrage, which can be modeled by various differential inequalitiesover the partial derivatives of the option price with respect to the expirationtime and the strike price. To address these challenges, we have introduced aDerivative-Constrained Neural Network (DCNN), which is an enhancement of amultilayer perceptron (MLP) that incorporates derivatives in the objectivefunction. DCNN allows us to generate a smooth surface and incorporate theno-arbitrage condition thanks to the derivative terms in the loss function. Innumerical experiments, we train the model using prices generated with the SABRmodel to produce smile and skewness parameters. We carry out different settingsto examine the stability of the calibrated model under different conditions.The results show that DCNNs improve the interpolation of the implied volatilitysurface with smile and skewness by integrating the computation of thederivatives, which are necessary and sufficient no-arbitrage conditions. Thedeveloped algorithm also offers practitioners an effective tool forunderstanding expected market dynamics and managing risk associated withvolatility smile and skewness.

No-Arbitrage Deep Calibration for Volatility Smile and Skewness