Syllabus for MATHGR5400_001_2025_3 - NON-LINEAR OPTION PRICING
Course Syllabus
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, the challenges faced by today’s practitioners mostly 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, credit valuation adjustment (CVA), transaction costs, illiquid markets, super-replication under delta and gamma constraints, etc.
We will strive to make this course reasonably comprehensive, and to find the right balance between ideas, mathematical theory, and numerical implementations. 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, kernel trick, etc. These techniques allow us to compute some quantities that are key ingredients of the nonlinear Monte Carlo algorithms.
The Python programming language will be used to provide simple numerical simulations illustrating the methods presented in the course. Homeworks will allow the students to check their understanding of the course by solving exercises inspired by our experience as quantitative analysts, and will involve some coding in Python.
Textbook
The main reference for this course will be the monograph Nonlinear Option Pricing [1] by Julien Guyon and Pierre Henry-Labordère.