Non linear pricing

Nonlinear option pricing

The classical curriculum of mathematical finance programs generally covers the link between linear parabolic partial differential equations (PDEs) and stochastic differential equations (SDEs), resulting from Feynman-Kac’s formula. However, today, many challenges faced by practitioners involve nonlinear PDEs. The aim of this course is to provide the students with the mathematical tools and numerical methods required to tackle these issues, and illustrate the methods with practical case studies like American option pricing, uncertain volatility, uncertain mortality, different rates for borrowing and lending, calibration of models to market smiles, and credit valuation adjustment (CVA).

We will spend some time on the theory: optimal stopping, stochastic control, backward stochastic differential equations (BSDEs), McKean SDEs, branching diffusions. But the main focus will deliberately be on ideas and numerical examples, which we believe help a lot in understanding the tools and building intuition.

PDE methods suffer from the curse of dimensionality. Since most quantitative finance problems are high-dimensional, we will mostly focus on simulation-based methods (a.k.a. Monte Carlo algorithms). This course exposes the students with a wide variety of Machine Learning techniques, old and new, including parametric regression, nonparametric regression, neural networks, etc. These techniques allow us to compute conditional expectations, which are key ingredients of the nonlinear Monte Carlo algorithms.

Textbook

The main reference for this course will be the monograph Nonlinear Option Pricing [1].

References

[1] Guyon, J. and Henry-Labordère, P.: Nonlinear Option Pricing, Chapman & Hall/CRC Financial Mathematics Series, 2014.