Distance Learning - Advanced Finite Difference Method (FDM) for Computational Finance - (code DL-FDM)

The goal of this distance learning course is to approximate the solution of partial differential equations (PDEs) by the Finite Difference Method (FDM) with applications to derivative pricing in computational finance. This course is an in-depth introduction from PDE model specification through efficient and accurate finite difference schemes for a range of one-factor and two-factor option pricing problems. The focus is on understanding the financial, mathematical and numerical skills needed in order to set up the discrete system of equations that we can then implement in C++11, for example.

This course is suitable for front-office and middle-office quant developers who wish to learn the finite difference method for computational finance. The contents of the course is also relevant to other disciplines such as science and engineering.

This course is given in several forms: as distance learning, at client site or as a regular event in various cities.

Subjects Covered

• Specifying financial models as PDEs. Choosing the ‘best’ PDE in a given use case.
• A range of modern finite difference schemes for one and two-factor problems.
• Supporting numerical methods (matrix algebra, nonlinear solvers, interpolators).
• C++11 and library integration.
• Assembling, running and testing the discrete system of equations.

What do you learn?

• The mathematical and numerical foundations of PDE and FDM.
• Applying finite difference methods to computational finance.
• Setting up algorithms and implementing them in a programming language.
• Running, testing and stress-testing the finite difference schemes.

 What do you receive?

• Do the course at your own pace. No time limit on course access.
• Copies of the books “Finite Difference Methods in Financial Engineering” and “Financial Instrument Pricing using C++, Second Edition” by Daniel J. Duffy.
• Ongoing support via email. Exercises, review and feedback.
• End-of-course exam (e.g. Skype) and Certificate.

We also provide a C++11 software framework (with full source code) that you can use in your work to test your schemes. Of course, you can use other languages such as Python or C#, for example.

Trainer and Originator

Daniel J. Duffy is the originator and mentor of this course. He has a PhD in the numerical analysis of partial differential equations from the University of Dublin (Trinity College).

Course contents updated October 2017

Course Contents

Part 1 Financial and Mathematical Models

In this section we discuss the financial and mathematical PDE models that we approximate using the finite difference method in later sections. We examine the properties of the solution in order to provide insights into the problem at hand. We discuss a number of methods such as the Fichera theory and domain transformation that we shall use in examples in computational finance.

PDE Categories

• One-factor, multi-factor
• Linear, semi-linear, non-linear
• Domain (bounded, semi-infinite, infinite)
• Time-dependent and time-independent PDEs
• Conservative and non-conservative PDE forms
• Reduction to first-order systems

Special Kinds of PDE

• Parabolic and elliptic PDE
• First-order hyperbolic PDE
• ‘Asian-style’ PDE
• Ordinary differential equations (ODEs)

Describing PDEs

• PDE coefficients
• Boundary conditions (Dirichlet, Neumann, none, linearity)
• Initial conditions
• Well-posedness and continuity
• Energy inequality; existence and uniqueness of solution

 Special Properties

• Convection dominance
• Discontinuous initial conditions
• Domain truncation and domain transformation
• Mixed derivatives
• The Fichera theory: Feller conditions

 Part 2 Finite Difference Method: Fundamental Techniques

We discuss one-factor PDEs in detail and their approximation by finite difference schemes. We apply these schemes to general convection-diffusion-reaction equations and boundary conditions and we show their application to the one-factor Black Scholes PDE.

We also discuss the numerical analysis of the finite difference method in which we give necessary and sufficient conditions for a finite difference scheme to be stable and to converge to the solution of the PDE that it is approximating. Some methods that we discuss are the Method of Lines (MOL), exponential fitting and the Alternating Direction Explicit (ADE) method.

Attention Points

• Continuous to discrete space: meshes and mesh generation
• Approximation of partial derivatives
• One-step and multistep time marching schemes
• Full discretisaton
• Semi-discretisation and Method of Lines (MOL)

Some Well-known Schemes

• Explicit and implicit Euler
• Crank Nicolson
• Richardson extrapolation
• Alternating Direction Explicit (ADE)
• Monotone schemes and M-matrices

Auxiliary Numerical Methods

• Solution of linear and nonlinear systems
• Interpolation and smoothing
• Numerical integration
• Optimisation (Levenberg-Marquardt, Differential Evolution)
• Eigen and Boost Matrix libraries

Analysis of FDM

• Stability, consistency and convergence
• Conditional and unconditional stability
• Von Neumann stability analysis
• Maximum principle
• Order of accuracy and rate of convergence

Example: One-Factor Black Scholes PDE, I

• Domain truncation versus domain transformation
• Call and put options: boundary conditions
• Payoff functions; handling discontinuities
• Crank Nicolson and Rannacher methods
• Using exponential fitting

Example: One-Factor Black Scholes PDE, II

• Avoiding oscillations: fully implicit method and extrapolation
• Critique of the Crank Nicolson method
• BDF2 and TR-BDF2 schemes
• ADE method for the Black Scholes PDE
• More general cases and Fichera boundary conditions
• Approximating the Greeks (sensitivities)

Numerical Tools

• LU and Cholesky decomposition
• Solution of tridiagonal matrix systems
• Linear and cubic spline interpolation
• Univariate and bivariate normal distributions

Part 3 Advanced (Nonlinear) Models

In this section we introduce a number of linear and nonlinear PDEs and finite difference schemes. In particular, we consider free and moving boundary values problems that describe an option’s early exercise features. Since this is a nonlinear problem we see that the methods from Part 2 are not directly applicable. We then resort to nonlinear solvers and transformations to make the problem more tractable.

We discuss the Method of Lines (MOL) in detail. This method reduces a PDE to a system of ordinary differential equations (ODEs) by discretising the underlying space variables only. The resulting ODE system can then be handed to a solver such as Mathematica’s NDSolve or the Boost C++ library odeint. These libraries are suitable for stiff and non-stiff systems of nonlinear ODEs.

Early Exercise Features

• Free and moving boundaries
• Formulations (fixed domain, front tracking)
• Variational inequalities and PSOR
• Brennan-Schwartz method
• Penalty methods

The Method of Lines (MOL) Overview

• Semi-discretisation
• Vertical MOL and horizontal MOL (Rothe’s method)
• Example: one-dimensional heat equation
• Advantages of MOL
• Application areas

MOL in Detail

• Stiff and non-stiff ODEs
• Linear and nonlinear systems
• Incorporating non-Dirichlet boundary conditions into MOL
• Adaptive and non-adaptive ODE solvers

MOL PDE Examples

• Black Scholes
• Cox Ingersoll Ross (CIR)
• Uncertain Volatility Model (UVM)
• CEV model
• Pde for credit value adjustment (CVA)
• MOL in Mathematica and Boost C++ odeint

ADE for one-Factor Problems

• Background and motivation
• Saul’yev, Barakat-Clark and Larkin variants
• ADE for convection terms
• Conditional consistency; stability
• Boundary conditions

 Other Differential Equations

• Fokker-Planck
• First-time exit PDE
• Riccati ODE

Kinds of Boundary Conditions

• Dirichlet, Neumann, Robin
• Linearity
• PDE on boundary (hyperbolic, parabolic)
• Fichera conditions

Part 4 Two-Factor Models

In this section we discuss several popular finite difference methods to approximate the solutions of the PDEs describing two-factor option pricing. We discuss Alternating Direction Implicit (ADI) method and the method of Fractional Steps (“Soviet Splitting”) which originated in the United States and the former Soviet Union in the 1960’s, respectively. We apply them to several PDEs in computational finance. Of particular importance is the problem of approximating the mixed derivatives in the PDE to ensure that the resulting scheme is monotone and does not lead to spurious oscillations.

We also discuss MOL and ADE for linear and nonlinear PDEs and we compare them with ADI and splitting methods.

Contenders

• Alternating Direction Implicit (ADI)
• Splitting (Fractional Steps method)
• ADE in two dimensions
• Other methods

The ADI Method

• Using ADI for two-factor PDE
• Mixed derivatives using Craig-Sneyd
• Test cases: basket options and Heston model
• Generalising the ADI method

The Operator Splitting Method

• Yanenko, Marchuk and Strang splittings
• Explicit and implicit splitting
• Handling mixed derivatives and boundary conditions
• Splitting and predictor-corrector methods
• Marchuk 1-2-2-1 model

The ADE Method

• Origins and background; how it differs from ADI and splitting
• Motivating ADE: from heat pde to convection-diffusion and mixed derivatives
• One-sided and centred variants of ADE
• ADE in 3 factors

Comparing ADI, Splitting and ADE Methods

• How they handle mixed derivatives
• Boundary conditions
• Accuracy and robustness of the schemes
• Improving accuracy
• Can the scheme be parallelised?

Mixed Derivatives

• Modeling correlation: extreme cases
• Craig-Sneyd, Verwer, Hout_Welfert, Yanenko
• Stress-testing mixed derivatives
• Test case: compare ADI, splitting and ADE for Heston model

Test Cases

• Basket options
• Heston model
• Asian options
• Anchoring model (Wilmott, Lewis and Duffy)
• Exact solutions

Modelling Jumps

• Merton’s and Kou models
• Partial Integro-Differential Equations (PIDE)
• Implicit-explicit Euler method
• Implicit-explicit Runge-Kutta method

Part 5 System Assembly, Software Framework and Stress Testing

In this final part we analyse finite difference schemes to determine their accuracy, efficiency and robustness for a range of input parameters. We need to investigate a number of numerical and computational attention points and we employ whatever tools and methods are needed in order to reach a conclusion. At this stage it may be desirable or even a requirement to view the project as having a software design aspect. For this course, it is an optimisation step but it is useful for front-office and middle-office quants who wish to create software systems from reusable software components.

We provide a number of software tools to promote the quality of the software testing process, including visualisation of results in the C++ Excel Driver interface, automatic testing by generating random numbers for parameter input and useful numerical tools such as the ability to interpolate values at non mesh points.

Basic Accuracy Testing (Proof-of-Concept)

• Estimating the local truncation error
• The effects of domain truncation and domain transformation
• Which boundary conditions are optimal?
• Choice of matrix solver

Improving Accuracy and Robustness

• Non-smooth payoffs
• Convection-dominance and exponential fitting
• Computing option sensitivities
• Automatic testing: producing a report

Testing Accuracy: Sparring Partners

• Is there an analytic/quasi-analytic solution to test against?
• Monte Carlo solution
• Binomial method solution
• Two-asset, Heston analytic solution

Using C++11 and C++14

• Random number and distributions library
• C++ Currency, parallel computing and tasks
• Loop unrolling
• Using OpenMP
• Boost and Eigen
  

C++ Software Framework

• Domain Architecture (Duffy)
• Object-oriented, generic and functional programming styles
• A defined and reproducible software architecture
• Variations; incremental development

Prerequisites

We assume basic knowledge of differential equations and finite difference theory. The models and examples in the course are taken from computational finance.

Some skills in arithmetical and algebraic manipulation are useful, especially when assembling systems of discrete equations.

Knowledge of a mathematical typesetting system (ideally, LaTeX) is strongly recommended.

Who should attend?

• Financial engineers who design new pricing models 
• Analysts and quants
  
• Other professionals who wish to understand and apply advanced numerical methods to derivatives pricing

Duration, price, date, locations and registration