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Core Processes

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. PDE Theory. Continuous problem classification Concentrate on 2 nd order parabolic equations

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Core Processes

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  1. 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. PDE Theory • Continuous problem classification • Concentrate on 2nd order parabolic equations • In fact, convection-diffusion equations • Existence, uniqueness and other qualitative properties

  3. 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. Examples • Standard Black Scholes PDE (different underlyings) • Barrier options • Asian options • Other path-dependent options • Several underlyings

  5. 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. 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. 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. 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. 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. 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. 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. Special Schemes (2/2) • Schemes for multidimensional problems • ‘Direct’ schemes • ADI (Alternating Direction Implicit) • Splitting methods (Janenko)

  13. Supporting Techniques (1/3) • Fourier series/transforms • Ordinary Differential Equations (ODE) • Stochastic Differential Equations (SDE) • Theory of PDE

  14. 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. 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

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