1 advanced finite difference methods for financial instrument pricing core processes pde theory in...
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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