Deep Local Volatility
Résumé
Deep learning for option pricing has emerged as a novel methodology for fast computations
with applications in calibration and computation of Greeks. However, many of these approaches
do not enforce any no-arbitrage conditions, and the subsequent local volatility surface is never
considered. In this article, we develop a deep learning approach for interpolation of European vanilla
option prices which jointly yields the full surface of local volatilities. We demonstrate the modification
of the loss function or the feed forward network architecture to enforce (hard constraints approach) or
favor (soft constraints approach) the no-arbitrage conditions and we specify the experimental design
parameters that are needed for adequate performance. A novel component is the use of the Dupire
formula to enforce bounds on the local volatility associated with option prices, during the network
fitting. Our methodology is benchmarked numerically on real datasets of DAX vanilla options.
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