  Mathematical modelling is ubiquitous in the financial industry and drives key
decision processes. Any given model provides only a crude approximation to
reality and the risk of using an inadequate model is hard to detect and
quantify. By contrast, modern data science techniques are opening the door to
more robust and data-driven model selection mechanisms. However, most machine
learning models are "black-boxes" as individual parameters do not have
meaningful interpretation. The aim of this paper is to combine the above
approaches achieving the best of both worlds. Combining neural networks with
risk models based on classical stochastic differential equations (SDEs), we
find robust bounds for prices of derivatives and the corresponding hedging
strategies while incorporating relevant market data. The resulting model called
neural SDE is an instantiation of generative models and is closely linked with
the theory of causal optimal transport. Neural SDEs allow consistent
calibration under both the risk-neutral and the real-world measures. Thus the
model can be used to simulate market scenarios needed for assessing risk
profiles and hedging strategies. We develop and analyse novel algorithms needed
for efficient use of neural SDEs. We validate our approach with numerical
experiments using both local and stochastic volatility models.
