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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

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PDE Theory

Continuous problem classification Concentrate on 2nd order parabolic

equations In fact, convection-diffusion equations Existence, uniqueness and other

qualitative properties

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gPDE 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

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Examples

Standard Black Scholes PDE (different underlyings)

Barrier options Asian options Other path-dependent options Several underlyings

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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

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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

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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

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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)

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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

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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

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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

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Special Schemes (2/2)

Schemes for multidimensional problems ‘Direct’ schemes ADI (Alternating Direction Implicit) Splitting methods (Janenko)

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Supporting Techniques (1/3)

Fourier series/transforms Ordinary Differential Equations (ODE) Stochastic Differential Equations (SDE) Theory of PDE

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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)

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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