Distance Learning - Advanced Finite Difference Method for Quantitative Finance: Theory, Applications and Computation - (code DL-FDM)

• Define an unambiguous, water-tight PDE for an option model
• Get a real understanding of finite differences, from A to Z
• Know which schemes work and when
• Apply FDM to a wide range of option pricing models
• Learn robust and accurate algorithms
• Guidelines on implementation in C++, C#, parallel and GPUs 

In short, you will learn the latest developments in this field and be able to use them immediately in your own work. Source code for the models is provided.

What do you receive?

• "My expectations were topped; can go now and implement 3d models using splitting"
• "Really liked it. Very informative"
• "I would enthusiastically recommend this course to colleagues"
• Excellent course

Course Contents

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

• 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

• PDE formulation
• Penalty method
• ADE with the Brennan Schwartz model
• Other methods (front fixing, front tracking, PVI)
  

• 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

• 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

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

• Asian options
• Basket options
• Heston model
• Black Scholes model with stochastic interest rates

• 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

• 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

• 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

• 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

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

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

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

• 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

• 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

• 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

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

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