Finite Difference Schemes with Exact Recovery of Vanilla Option Prices

Finite difference schemes are the preferred tool for solving the partial differential equations that arise when pricing exotic derivatives. The schemes available up to now can achieve at most O(∆t²) accuracy in time and O(∆x²) accuracy in space when recovering instrument prices. We show how to discretize finite difference solvers to exactly recover the prices of all vanilla options on the grid. Our approach can be used with any choice of grid, and leads to a specific formula for discretization of Dupire's local volatility. It can also be applied to local stochastic volatility and interest rate hybrid models