Finite-difference Numerical Methods of Partial Differential Equations in Finance with Matlab

This is the main aim of this course.
Here you can find the notes of this course and below you have the videos with all the explanations.

1. Ordinary Differential Equations (ODEs):

Numerical resolution of first order ODEs:

Forward Euler method.

Matlab program with the Forward Euler method, (forward_euler.m).

Matlab program with the Midpoint method, (midpoint.m).

Errors and stability in numerical methods.

Stability of forward and backward Euler methods.

2. Finite-difference methods to solve second-order partial differential equations (PDEs):

Stability Analysis Fourier Approach von Neuman.

The advection or wave equation.

Explicit forward time centred space method advection equation.

The Lax method.

Staggered leapfrog method.

The heat or diffusion equation.

Explicit forward time-centred space method for the diffusion equation.

Fully implicit method for the diffusion equation.

Crank-Nicholson method for the diffusion equation.

Inverting matrices more efficiently:

3. Finite-difference methods to solve the Black-Scholes equation:

Introducing the Black-Scholes equation:

Finite-difference representations for the Black-Scholes equation.

Explicit FTCS method for the Black-Scholes equation.

Fully implicit method for the Black-Scholes equation.

Matrix representation of the fully implicit method for the Black-Scholes equation.

The Crank-Nicholson method for the Black-Scholes equation.

Matrix representation of the Crank-Nicholson method for the Black-Scholes equation.

Free boundary problems: American options.

Bibliography:

P. Wilmott, S. Howison and J. Dewinne, "The Mathematics of Financial Derivatives", Cambridge University Press, 1976.
D. Tavella and C. Randall, "Pricing Financial Instruments", John Willey Sons Inc., 2000.
G.D. Smith, "Numerical Solution of Partial Differential Quations: Finite Difference Methods", Clarendon Press, Oxford, 1985.
J.C. Strikwerda, "Finite Difference Schemes and Partial Differential Equations", Chapman and Hall, New York, 1990.
J.C. Hull, "Options, Futures and other derivatives", Prentice Hall, 2000.