  Volatility smile and skewness are two key properties of option prices that
are represented by the implied volatility (IV) surface. However, IV surface
calibration through nonlinear interpolation is a complex problem due to several
factors, including limited input data, low liquidity, and noise. Additionally,
the calibrated surface must obey the fundamental financial principle of the
absence of arbitrage, which can be modeled by various differential inequalities
over the partial derivatives of the option price with respect to the expiration
time and the strike price. To address these challenges, we have introduced a
Derivative-Constrained Neural Network (DCNN), which is an enhancement of a
multilayer perceptron (MLP) that incorporates derivatives in the objective
function. DCNN allows us to generate a smooth surface and incorporate the
no-arbitrage condition thanks to the derivative terms in the loss function. In
numerical experiments, we train the model using prices generated with the SABR
model to produce smile and skewness parameters. We carry out different settings
to examine the stability of the calibrated model under different conditions.
The results show that DCNNs improve the interpolation of the implied volatility
surface with smile and skewness by integrating the computation of the
derivatives, which are necessary and sufficient no-arbitrage conditions. The
developed algorithm also offers practitioners an effective tool for
understanding expected market dynamics and managing risk associated with
volatility smile and skewness.
