solving the Black-Scholes PDE by finite differences
(Add diagram of domain here…)
0.1 Finite-difference formulae
We summarize the equations for the finite differences below.
Let , , be some chosen positive integers, which determine the grid on which we are approximating the solution of the PDE.
Set to be the approximation to , for and . (For convenience, we have made time move “backwards” as we increase , because the original PDE is really a backwards heat equation, and evolves backwards in time.)
Since the PDE to solve is parabolic and time-dependent, we can step through time to numerically approximate it. Given , we can recursively compute .
(Add stencil of numerical method here…)
0.2 Convergence of methods
(Briefly discuss convergence properties of these methods here…)
0.3 Example results
Boundary conditions and parameters:
(Describe analytic solution here…)
http://svn.gold-saucer.org/math/PlanetMath/SolvingTheBlackScholesPDEByFiniteDifferences/bss.pyPython program that implements the finite-difference methods for the above two problems, and plots the results
|Title||solving the Black-Scholes PDE by finite differences|
|Date of creation||2013-03-22 16:30:59|
|Last modified on||2013-03-22 16:30:59|
|Last modified by||stevecheng (10074)|