Distance Learning - PDE/FDM Methods in Computational Finance: Theory, Algorithms and Applications - (code DL-PDEFDM)
The goal of this distance learning course is to introduce the finite difference method and its applications to computational finance. We bring together in one place all the methods and techniques that are used to price and hedge derivatives that are modelled using partial differential equations (PDE) in finance and that we approximate using the Finite Difference Method (FDM). We discuss all aspects of the problem, from PDE definition to numerical schemes and system assembly.
This course should appeal to a wide audience in finance. Having followed this course you will be in a position to understand the literature on PDE/FDM methods. It also lays the foundation for more advanced methods for multi-factor and nonlinear models that are used in computational finance.
- A complete overview A-Z of PDE/FDM for one-factor models in finance.
- The main FDM methods: when to use.
- State-of-the-art, innovative and high-performance FDM methods.
- How to design and implement FDM schemes.
- To-the-point intensive overview of PDE/FDM.
- One-stop: all relevant results assembled in one place.
- Understand the literature on PDE in finance.
- Test the schemes by running the C++/C# code that you receive as part of the course.
- Follow-up: projects and more advanced PDE/FDM topics.
University and Company Cooperation
Site licences are offered to both universities and banks. If you have several colleagues or students who wish to partake in this course please contact (+31-72-2204802).
Clients and References
We train quants and traders around the world using distance learning and on-site training. We also train MFE students and supervise their projects at several universities. For references and testimonials please contact (+31-72-2204802).
About the Trainer / Course Creator
Dr. Daniel J. Duffy has MSc and PhD degrees in Numerical Analysis from Trinity College, Dublin. He has applied numerical methods in engineering and computational finance, in particular the application of exponentially fitted methods, Splitting and Alternating Direction Explicit (ADE) methods to derivatives pricing problems.
You need to purchase the book Finite Difference Methods in Financial Engineering: A Partial Differential Equation Approach by Daniel J. Duffy ISBN: 978-0-470-85882-0 to accompany this course.
He trains professionals in finance and MFE/MSc students at several universities.
What do you get?
Approximately 50 hours audio/videos explaining the finite difference for computational finance, detailed descriptions of the numerical algorithms, C++/C# code and a software framework that you can extend to test your schemes.
Part 1 - Partial Differential Equation (PDE) Theory
- Classification of partial differential equations
- Elliptic, parabolic and hyperbolic equations
- Linear and nonlinear PDE
- Initial value and initial boundary value problems for PDE
Initial Examples of PDE
- Laplace and Poisson equations
- Heat equation
- Fokker-Planck and Kolmogorov equations
PDE in Computational Finance
- PDE with degenerate characteristic form
- The Fichera treatment of boundary conditions
- Mixed derivatives
- Boundary conditions (Dirichlet, Neumann, Robin, Linearity)
- Classification of PDE in computational finance
- Smoothing of payoff function (initial condition)
- Domain truncation versus domain transformation
- Convection dominance and small volatility
- Conservative form versus non-conservative form PDEs
Part 2 - The Finite Difference Method (FDM) in a Nutshell
Overview of FDM for PDE
- Meshes and divided differences
- Discretisation of PDE and truncation error
- Notation and assembly of FD schemes
- FDM for elliptic, parabolic and hyperbolic PDE
Analysis of Finite Difference Schemes : Theory
- Initial value and initial boundary value problems
- Consistency (unconditional and conditional variants)
- Stability (unconditional and conditional variants)
- Convergence and order of convergence
Analysis of Finite Difference Schemes : Methods
- Fourier analysis of FD schemes
- Monotone methods and M-matrices
- Matrix eigenvalue analysis
Model Problems and their Approximation
- Rationale and motivation
- Two-point boundary value problem
- Heat Equation
- Convection (advection) equation
- Convection-diffusion equation
Part 3 - FDM for One-factor Finance Models
- Classification of one-factor PDEs
- Essential difficulties and their solutions
- Choice of schemes
- Comparing schemes
One-Factor European Options
- Crank Nicolson, Euler and ADE schemes
- Assembly of matrix system; LU decomposition
- Discontinuous payoff functions
- Domain truncation versus domain transformation
- Avoiding oscillations and static instability
- Computing the greeks
- Formulation of PDE
- Single barriers, double barriers and exponential barriers
- Continuous and discrete monitoring
- Comparing Crank Nicolson and ADE
Early Exercise and American Options
- Problem formulations: how many?
- Penalty method
- PSOR (Projected SOR)
FDM for Interest Rate Models
- Caps and floors
- Vasicek, CIR, HW
- Bond and Bond options
- Convertible bonds
Part 4 - Advanced FDM in Computational Finance
An Introduction to Alternating Direction Explicit (ADE)
- Rationale, background and motivation
- Example 101: Two-factor heat equation
- ADE for convection-diffusion PDE (the variants)
- ADE for one-factor PDEs in finance
- Dupire’s equation and calibration
- ADE for nonlinear problems (e.g. Uncertain Volatility)
Improving Robustness of FD Schemes
- Smoothing of discontinuous payoff functions; the choices
- Using domain transformation versus of domain truncation
- Exponential fitting for convection-dominated PDE
- Monotone schemes and ensuring positive solutions
FDM versus Lattice Methods
- Applicability to one-factor and n-factor models
- Early exercise; computing the greeks
- Stability and accuracy comparisons
- Ease of use and implementation
Preview of Advanced Models
- Alternating Direction Implicit (ADI)
- Soviet Splitting methods
- ADE in multi-dimensions
- Partial Integro-Differential Equations (PIDE)
The Computational Process
- Choice of space and time discretisers
- Approximation of PDE coefficients, boundary and initial conditions
- Assembling the system of equations
- Testing and debugging
Knowledge of calculus is assumed, for example partial derivatives and Taylor’s theorem. Familiarity with matrix algebra is also advantageous.
Who should attend?
This course is for a broad group of professionals in finance: quant developers and analysts, model validators and support teams. In particular, this course is for those who may not have had exposure to PDE/FDM during their university education and who wish to become acquainted with PDE methods.
This course is also of interest to the academic community and those engineering and IT personnel wishing to make the transition to finance.
Duration, price, date, locations and registration
You study in your own pace. Under normal circumstances, this should take you between 1 and 1.5 years to complete.
|Dates and location:
||(click on dates to print registration form)
Click here to register.
This course is meant for two main groups: first, full-time students at accredited universities and places of learning and second those professionals working in finance, industry and business.
The prices are:
- Full-time students outside the European Union: Euro 399
- Full-time students inside the European Union: Euro 482,79 (including 21% VAT)
- For those working in finance/industry: Euro 899 excluding VAT.
For private persons/students inside the EU we need to calculate 21% VAT. Companies inside the EU need to provide a VAT number, else VAT will be calculated too. For companies and private persons situated outside the EU, no VAT will be charged.
Payment is by means of (electronic) bank transfer. We do not accept checks or credit-cards.
Because of the special student price, students must supply evidence that shows you are a full-time student and will need to supply a university email address in order to login to the audio shows and forums.
You can register by clicking the registration link above ("Duration, price, date, locations and registration" section). This will bring up the registration form which you need to fill in. After submitting the registration form, you will receive a reservation e-mail. You will need to print the reservation e-mail, sign it and send it back to us. This because we need a signature.