MoneyScience Financial Training

Advanced Finite Difference Method for Quantitative Finance: Theory, Applications and Computation
Location: London, UK    |   Tutor: Daniel Duffy
Course Length: 3 Days   |   Cost: £3000 

Part 1 - One-Factor Models

General Considerations
- Approximating derivatives by divided differences
- Specifying boundary conditions
- Payoff functions, monitoring points
- Solving the system of equations

Choice of Time Discretization
- Euler, Crank-Nicolson, Rannacher methods
- Alternating Direction Explicit (ADE) method
- Using Bulirsch-Stoer for stiff equations
- Richardson extrapolation

Special Attention Areas and their Resolution
- Discontinuous coefficients
- Extreme (large, small) volatility and drift terms
- Avoiding oscillations in the greeks (for example, at the strike price)
- Spikes in barrier options
- Adaptive meshing

Early Exercise Features
- PDE formulation
- Penalty method
- ADE with the Brennan Schwartz model
- Other methods (front fixing, front tracking, PVI)

Test Cases
- European vanilla option: domain truncation versus domain transformation
- Barrier options and exponential fitting; time-dependent barriers
- American options using the Brennan-Schwartz algorithm
- Robust approximation to the greeks

The Keller Box Scheme
- Reduction of Black Scholes PDE to first-order system
- Accomodating discontinuous payoffs and boundary conditions
- Formulating the box scheme as first-order system
- Achieving second-order accuracy in price and delta

Special Topics
- Mollifiers and smoothing of payoff functions
- Conservative versus non-conservative PDE forms; which one?
- Modelling transity density and Fokker-Planck equation
- Calibration


Part 2 - Two-Factor Models

Some Partial Differential Equations for two Factors
- Asian options
- Basket options
- Heston model
- Black Scholes model with stochastic interest rates

Analysing Multi-factor PDEs
- The Fichera theory and non-negative characteristic forms in finance PDEs
- Using Fichera theory to deduce and discover boundary conditions
- The relationship between Fichera function and the Feller condition for Heston and CIR models
- Fichera function and domain transformations

The ADI Method
- Using ADI for two-factor PDE
- Mixed derivatives using Craig-Sneyd and Hout/Welfert
- 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
- Example: baskets, Heston, SABR PDEs

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 and more factors
- ADE and how it is parallelised

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 parallelized?

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

Advanced Techniques
- Coordinate and domain transformation
- Adaptive meshing
- Mixed PDE-Monte Carlo method
- Random walk in a PDE
- Transition density computation using ADE

Part 3 - Three-Factor, Hybrid and Interest Rate Applications

An Introduction to Three-Factor Modelling
- Model: the 3d Heat equation and Poisson’s equation
- Comparing ADE and splitting methods
- Creating a reusable baseline software structure for computational finance
- Parallelisation and speedup

Modelling Jumps
- Merton’s and Kou model
- Partial Integro-Differential Equations (PIDE)
- Finite Difference Methods for PIDE
- An Introduction to the Finite Element Method (FEM) for PIDE

Interest-rate Models and PDE/FDM
- One-factor models (Vasicek, CIR, Hull-White)
- Comparing domain truncation and domain transformation
- Recovering the Feller condition
- Problem schemes; challenges and solutions
- Using the ADE method for one-factor IR models

Three-Factor PDEs and FDM Solutions
- Multi-asset (basket) options
- Hull-White-Heston model
- Numerical solution of 3-factor models
- Multiple mixed derivative terms
- Libraries, design and object-oriented solutions