Advanced Numerical Methods (2019-2020)

2019-2020
Lecturer(s): 
Prof. Christoph Reisinger
Course Term: 
Hilary
Course Overview: 

This course covers some advanced numerical methods for a selection of important computational problems in finance, such as American options (obstacle problems), PDEs for multi-factor models (eg, from stochastic volatility), and the calibration of volatility models to quoted market prices. At the end of the course, the student should have a thorough understanding of the theory behind more advanced finite difference and calibration methods, be able to implement them for a range of applications, and have an appreciation of some of the current research areas.

Course Syllabus: 

Lectures 1-4:
American options: explicit treatment, projected iterations, penalty and policy iteration methods; multi-dimensional PDEs: discretisation of cross-derivatives, alternating direction implicit (ADI) schemes.

Lectures 5-8:
Calibration of parametric models: example of the Heston model, non-linear constrained optimisation, practical issues; calibration of non-parametric models: local volatility, Dupire's formula, ill-posed inverse problems and link to model uncertainty; (Tikhonov) regularisation of ill-posed problems, Bayesian inverse problems; calibration of local-stochastic volatility models, particle methods.

Reading List: 

K. in 't Hout, Numerical Partial Differential Equations in Finance Explained, Palgrave, 2017.
A. Hirsa, Computational Methods in Finance, Chapman & Hall/CRC, 2012.
J. Guyon, P. Henry-Labordere, Nonlinear Option Pricing, Chapman & Hall/CRC, 2014.

Please note that e-book versions of many books in the reading lists can be found on SOLO and ORLO.