1. 1 Advanced Finite Difference Methods for Financial Instrument Pricing Core Processes PDE theory in general Applications to financial engineering State of the art finite difference theory Applying FDM to financial engineering Algorithms and mapping to code

2. 2 Advanced Finite Difference Methods for Financial Instrument Pricing PDE Theory Continuous problem classification Concentrate on 2 nd order parabolic equations In fact, convection-diffusion equations Existence, uniqueness and other qualitative properties

3. 3 Advanced Finite Difference Methods for Financial Instrument Pricing PDE and Financial Engineering Derive PDE from Ito’s lemma Applicable to a range of plain and exotic option problems Document and standardise option classes Concentrate on one-factor and two-factor models

4. 4 Advanced Finite Difference Methods for Financial Instrument Pricing Examples Standard Black Scholes PDE (different underlyings) Barrier options Asian options Other path-dependent options Several underlyings

5. 5 Advanced Finite Difference Methods for Financial Instrument Pricing Other PDE Types First-order hyperbolic equations ‘Mixed’ (e.g. parabolic/hyperbolic) Systems of equations Parabolic Integral Differential Equations (PIDE) Free and moving boundary value problems

6. 6 Advanced Finite Difference Methods for Financial Instrument Pricing Finite Difference Methods Lots of choices! Choosing the most appropriate one demands insight and experience Using a numerical recipe approach does not always work well This course resolves some of the problems and misunderstandings

7. 7 Advanced Finite Difference Methods for Financial Instrument Pricing FDM Types (1/2) Standard recipes (e.g. Crank Nicolson) Special FDM for difficult problems FDM for multi-dimensional problems FDM and parabolic variational inequalities

8. 8 Advanced Finite Difference Methods for Financial Instrument Pricing Standard FDM Crank Nicolson The standard FDM for many problems Does not work well in all cases This course tells why (and how to resolve the problem)

9. 9 Advanced Finite Difference Methods for Financial Instrument Pricing Special FDM Needed when we wish to address some difficult issues Standard recipes need to be replaced in thee cases We must defend why we need thee new schemes

10. 10 Advanced Finite Difference Methods for Financial Instrument Pricing What are the Problems? Producing stable and oscillation-free schemes Approximating the Greeks How to solve multi-dimensional problems Modeling free boundaries and the American exercise variation

11. 11 Advanced Finite Difference Methods for Financial Instrument Pricing Special Schemes (1/2) Fitted schemes Oscillation-free scheme The Box scheme (modelling Black Scholes as a first-order systems) Van Leer and other nonlinear schemes

12. 12 Advanced Finite Difference Methods for Financial Instrument Pricing Special Schemes (2/2) Schemes for multidimensional problems ‘Direct’ schemes ADI (Alternating Direction Implicit) Splitting methods (Janenko)

13. 13 Advanced Finite Difference Methods for Financial Instrument Pricing Supporting Techniques (1/3) Fourier series/transforms Ordinary Differential Equations (ODE) Stochastic Differential Equations (SDE) Theory of PDE

14. 14 Advanced Finite Difference Methods for Financial Instrument Pricing Supporting Techniques (2/3) Finite differences for ODE and SDE Method Of Lines (MOL) and semi- discretisation Spectral and pseudospectral methods (Finite Element/Volume methods)

15. 15 Advanced Finite Difference Methods for Financial Instrument Pricing Supporting Techniques (3/3) Numerical differentiation, integration and interpolation Matrix algebra; solution of linear equations Discrete methods for PVI Monte Carlo, binomial and trinomial methods