Stochastic volatility models, where the volatility is a stochastic process, can capture most of the essential stylized facts of implied volatility surfaces and give more realistic dynamics of the volatility smile or skew. However, they come with the significant issue that they take too long to calibrate. Alternative calibration methods based on Deep Learning (DL) techniques have been recently used to build fast and accurate solutions to the calibration problem. Huge and Savine developed a Differential Deep Learning (DDL) approach, where Machine Learning models are trained on samples of not only features and labels but also differentials of labels to features. The present work aims to apply the DDL technique to price vanilla European options (i.e. the calibration instruments), more specifically, puts when the underlying asset follows a Heston model and then calibrate the model on the trained network. DDL allows for fast training and accurate pricing. The trained neural network dramatically reduces Heston calibration's computation time. In this work, we also introduce different regularisation techniques, and we apply them notably in the case of the DDL. We compare their performance in reducing overfitting and improving the generalisation error. The DDL performance is also compared to the classical DL (without differentiation) one in the case of Feed-Forward Neural Networks. We show that the DDL outperforms the DL.

翻译：随机波动率模型（其中波动率为随机过程）能够捕捉隐含波动率曲面的大部分典型特征，并提供波动率微笑或偏斜的更真实动态。然而，这类模型面临一个显著问题：校准过程耗时过长。基于深度学习技术的替代校准方法近年来被用于构建快速准确的校准解决方案。Huge与Savine提出了微分深度学习方法，该方法训练机器学习模型时不仅使用特征和标签样本，还利用标签对特征的微分。本研究旨在将DDL技术应用于欧式香草期权（即校准工具）的定价，具体为标的资产服从Heston模型时的看跌期权定价，并基于训练好的网络对模型进行校准。DDL方法可实现快速训练与精准定价，训练后的神经网络能大幅缩短Heston模型的校准计算时间。本文还引入了多种正则化技术，并特别将其应用于DDL场景，比较了它们在减少过拟合和改善泛化误差方面的表现。在前馈神经网络框架下，我们进一步对比了DDL与传统深度学习（无微分）的性能，结果表明DDL优于传统DL。