Machine learning for derivatives

V. Lemaire

The course illustrates the algorithmic techniques of Machine Learning used recently in Finance, and more precisely the approximation by neural networks. We introduce the Feedforward Neural Network (multilayer perceptron) and the universal appoximation theorem in $L^2$. We briefly recall the concept of supervised learning and the particularities when using simulated data. The learning algorithm used is Adam’s algorithm which is briefly explained. This very applied course is accompanied by codes in PyTorch.

The first application is the pricing of American options. The theory of discrete-time optimal stopping (Snell envelope) is presented and the Monte Carlo Regression technique (Longstaff-Schwartz algorithm) widely used to solve optimal stopping problems is introduced. The extension of this algorithm to neural networks can be done in different ways and we consider the two approaches from the papers:

  • Becker, Sebastian, Patrick Cheridito, and Arnulf Jentzen. “Deep Optimal Stopping.” Journal of Machine Learning Research 20, no. 74 (2019): 1-25.
  • Lapeyre, Bernard, and Jérôme Lelong. “Neural Network Regression for Bermudan Option Pricing.” Monte Carlo Methods and Applications 27, no. 3 (September 1, 2021): 227-47.

The second application is the hedging of European options. We recall the so-called delta-neutral hedge, the mean-quadratic hedging problem and the generalization to more general risk measures. The article that serves as a basis for this part of the course is:

Assessment is by:

  • weekly individual assignments in `PyTorch'
  • a final exam (written).

This course is not open to distance learning.