Overview - 2023.2 English

Finite Difference Methods are a family of numerical techniques to solve partial differential equations (PDEs). They do this by discretizing the continuous equation in the spatial dimensions (forming a multidimensional grid), and then iteratively evolving the system over a series of N discrete time steps. The result is a discrete solution for each individual grid point.

In the provided solver, the PDE is the Heston Pricing Model [HESTON1993]. Using the naming conventions of Hout & Foulon [HOUT2010], the PDE is given as:

\[\frac{\partial u}{\partial t} = \tfrac{1}{2}s^2v\frac{\partial^{2} u}{\partial s^2} + \rho\sigma sv\frac{\partial^{2} u}{\partial s\partial v} + \tfrac{1}{2}\sigma^2v\frac{\partial^{2} u}{\partial v^2} + (r_d - r_f)s\frac{\partial u}{\partial s} + \kappa(\eta - v)\frac{\partial u}{\partial v} - r_d u\]

Where:

\(u\) - the price of the European option;

\(s\) - the underlying price of the asset;

\(v\) - the volatility of the underlying price;

\(\sigma\) - the volatility of the volatility;

\(\rho\) - the correlation of Weiner processes;

\(\kappa\) - the mean-reversion rate;

\(\eta\) - the long term mean.

The Heston PDE then describes the evolution of an option price over time (\(t\)) and a solution of this PDE results in the specific option price for an \((s,v)\) pair for a given maturity date \(T\). The finite difference solver maps the \((s,v)\) pair onto a 2D discrete grid, and solves for option price \(u(s,v)\) after \(N\) time-steps.