These notes will cover some common methods used to handle discontinuities in payoffs when using a PDE pricer.

Convergence remedies

When saying convergence remedies I mean methods that we can employ to re-obtain the expected rate of convergence that is otherwise lost when the payoff is not smooth.

Averaging cells of Initial Conditions

In this method, nodal values are replaced with averages of surrounding values in the form:

\[ f_i = \frac{1}{S_{i+\frac12} - S_{i-\frac12}} \int_{S_{i-\frac12}}^{S_{i+\frac12}} f(y)\,dy \]

where \(f\) denotes an option's payoff

digital example?

gauss3 rule (3-point Gauss–Legendre quadrature rule)

Shifting The Mesh (not (yet?) implemented)

uniform

non-uniform / clustered

Projecting The Initial Conditions

references:

Pooley, D. M., Vetzal, K. R., & Forsyth, P. A. (2002, June 17). Convergence remedies for non-smooth payoffs in option pricing. University of Waterloo.